Area rule — Whitcomb's rule for transonic drag reduction: Question bank

3.1.26 · D5 · Physics › Compressible Flow & Aerodynamics › Area rule — Whitcomb's rule for transonic drag reduction

Recall Is bank ko kaise use karein

Har line ek prompt ::: answer hai. Answer ko cover karo, zor se bol ke commit karo, phir reveal karo. Answers sirf verdict nahi, reasoning dete hain — aur wahi reasoning asli point hai. Ye bank area rule aur uske neighbouring concepts ke specific traps ko target karta hai: Sears–Haack body, Transonic flow, Shock waves and wave drag, Slender-body theory, Mach angle and Mach cone, Prandtl–Glauert and compressibility corrections, aur drag breakdown.

True ya false — justify karo

True ya false: Agar do aircraft ka area distribution

A (x)

identical hai toh unka transonic wave drag bhi identical hoga, chahe ek sirf fuselage ho aur doosra fuselage-plus-wings.

True. von Kármán–Sears integral sirf

A (x)

par depend karta hai, isliye components ka split wave drag ko dikhta hi nahi — wahi equivalence hai area rule.

True ya false: Area ruling aircraft ka total drag har speed par kam karta hai.

False. Ye sirf wave drag katta hai, jo Mach 1 ke aas-paas aur usse upar exist karta hai. Skin-friction aur induced drag untouched rehte hain, aur low subsonic speeds par pinched waist friction/structural cost thoda badha bhi sakta hai.

True ya false: Ek perfectly cylindrical body of revolution ka transonic wave drag zero hota hai.

"Zero" ke baare mein False, "small" ke baare mein True. Constant-area middle mein

A^{''} = 0

hota hai, lekin nose aur tail ko area khol aur band karna hi padta hai, aur unhi regions mein wave drag hota hai — koi closed body aisa nahi ho sakta jisme

A^{''} \equiv 0

har jagah ho.

True ya false: Area rule ek statement hai is baare mein ki aircraft mein kitna area hai.

False. Ye is baare mein hai ki area kaise distribute hai aur kitni smoothly change karta hai (

A^{''}

), total amount ke baare mein nahi. Ek bada lekin gently varying

A (x)

ek chhote lekin kinked one ko beat kar sakta hai.

True ya false: Sears–Haack body wo shape hai jisme kisi bhi tarah ka drag sabse kam ho.

False. Ye sirf fixed length aur volume ke liye linearized supersonic theory mein wave drag minimize karta hai; ye friction ya induced drag ke baare mein kuch nahi kehta, aur ye slenderness assume karta hai.

True ya false: Exactly Mach 1 par tum aircraft ko flow ke perpendicular planes se kaatte ho, lekin Mach 2 par tilted Mach planes use karte ho.

True.

M \approx 1

ke paas Mach angle

μ = arcsin (1/ M) \to 9 0^{\circ}

hota hai, isliye Mach plane essentially perpendicular hota hai; jaise

M

badhta hai

μ

shrink hota hai aur cutting plane tilt ho jaati hai (Jones supersonic area rule).

True ya false: Wings ko thinner banana area rule satisfy karne ke liye kaafi hai.

False. Thinner wings apna area bump shrink karte hain lekin

A_{tot} (x)

mein bump phir bhi rehta hai; tumhe phir bhi fuselage pinch karna hoga taaki sum smooth ho. Kisi ek part ki smoothness nahi, total ki smoothness matter karti hai.

True ya false:

A (x)

curve mein ek kink (corner) same height ke smooth hump se worse hota hai.

True. Ek kink

A^{'}

ko jump karwata hai, isliye

A^{''}

ek impulse spike ban jaata hai, aur von Kármán integral bade concentrated

A^{''}

ko heavily penalize karta hai.

Error dhundho

D_{w} \propto V^{2} / ℓ^{4}

, isliye main volume ko nearly zero tak shrink karke wave drag jitna chaahun utna chhota kar sakta hun."

Error ye hai: tum woh payload/fuel bhi kho dete ho jiski wajah se volume tha. Ye scaling ek design trade hai fixed useful volume par — asli lever length badhana (

ℓ^{4}

) hai, aircraft ke existence ka reason delete karna nahi.

"Wave drag

d A / d x

ke large hone se aata hai, isliye steep sides wala body problem hai."

Error ye hai: driver

A^{''} = d^{2} A / d x^{2}

hai (curvature / slope ka rate of change),

A^{'}

nahi. Ek steep lekin straight ramp ka wahan

A^{''}

zero hota hai; shocks slope ki abrupt change se aate hain.

"Coke-bottle waist fuselage volume reduce karke kaam karta hai, jo drag reduce karta hai."

Error ye hai: waist is liye kaam karta hai ki fuselage area exactly wahan dip karta hai jahan wing area peak karta hai, taaki dono cancel ho jayein aur

A_{tot} (x)

smooth rahe. Mechanism curvature ka cancellation hai, volume ka removal nahi.

"Kyunki area ruling ne F-102 ki help ki, har aircraft ko wasp waist hona chahiye."

Error ye hai: benefit sirf transonic/supersonic regime tak confined hai. Ek purely subsonic aircraft ko kuch nahi milta aur friction/structure penalty bhugtani padti hai, isliye waist tab hi worth it hai jab tum actually Mach 1 cross karo.

"Source strength

f = U_{\infty} A (x)

hai, kyunki source ko poora cross-section fill karna hota hai."

Error ye hai: source strength

f = U_{\infty} d A / d x

hai. Streamtube ko sirf increment

d A

over

d x

ke liye volume shed karna hota hai — constant-area section ko kisi source ki zaroorat hi nahi.

"Kyunki close points ke liye

ln ∣ x_{1} - x_{2} ∣

negative hota hai, double integral

D_{w}

ko negative bana sakta hai."

Error ye hai: saamne ka minus sign aur

A^{''} A^{''}

self-correlation structure milke guarantee karte hain ki

D_{w} \geq 0

(drag negative nahi ho sakta). Kernel ka sign ek positive-definite form ke andar bookkeeping hai, thrust ka hint nahi.

Why questions

Flow ko "care kyun nahi" ki area wing se aayi ya fuselage se?

Slender-body theory mein disturbance axis par strength

U_{\infty} d A / d x

ki sources ki ek line hai; source sirf total

d A / d x

jaanta hai jo usse accommodate karni hai, us geometry se andha jisne use produce kiya.

Second derivative

A^{''}

kyun control karta hai wave drag ko,

A

A^{'}

kyun nahi?

A^{''}

measure karta hai ki area ka slope kitni abruptly change karta hai; abrupt changes hawa ko sharply accelerate/decelerate karne par majboor karti hain, aur yahi wo shocks launch karta hai jo momentum ko drag ke roop mein le jaate hain.

Same volume ka longer body itna kam wave drag kyun rakhta hai?

Same volume ko zyada length par spread karne se

A (x)

flat ho jaata hai, har jagah

A^{''}

shrink hota hai; effect double integral ke through

1/ ℓ^{4}

scaling mein compound hota hai — length sabse bada available lever hai.

Supersonic speeds par cutting plane perpendicular rehne ki bajaye kyun tilt karta hai?

Disturbances Mach cones ke along angle

μ = arcsin (1/ M)

par propagate karte hain; ek point sirf body ke unhi parts ko "feel" karta hai jo uski Mach cone par hain, isliye relevant cross-section Mach plane hai, vertical nahi.

Sears–Haack body dono ends par pointed (zero area) kyun hai, blunt kyun nahi?

A (x)

aur uski curvature ko chhota aur continuous rakhne ke liye, area ko smoothly zero se khulna aur smoothly zero par band hona chahiye; ek blunt end

A

ko finite value par le aata,

A^{''}

aur drag spike kar jaata.

Area ruling transonic story se kyun belong karta hai, subsonic se nahi?

Wave drag ke liye shock waves chahiye hote hain, jo tabhi appear hote hain jab local flow Mach 1 reach kare; usse neeche koi shock nahi, isliye koi wave drag nahi jise area rule attack kare.

Edge cases

Edge case: Fuselage jahan wing root se mila aur area curve mein sharp corner ban gaya, wahan

A^{''}

kya hai, aur ye kyun matter karta hai?

A^{''}

corner par ek Dirac impulse (infinite spike) ban jaata hai; drag integral mein daalne par ye bada contribution deta hai, isliye real designs junction ko smooth curve mein fair karte hain.

Edge case: Jab

M \to 1^{+}

, Mach angle aur tilted cutting plane ka kya hota hai?

μ = arcsin (1/ M) \to 9 0^{\circ}

, isliye Mach plane rotate karke perpendicular ho jaata hai — supersonic (Jones) area rule smoothly transonic (Whitcomb) perpendicular-plane rule ko apni limit ke roop mein recover kar leta hai.

Edge case: Area rule kya predict karta hai aise wing-body ke liye jiska

A (x)

already fuselage pinching ke bina ek Sears–Haack curve hai?

Koi waist ki zaroorat nahi — ye already optimal hai. Coke-bottle shape sirf fix hai jab raw sum bumpy ho; jo configuration apne aap smooth hai usse pinching se kuch nahi milta.

Edge case: Ek body ka

A (x)

perfectly smooth hai lekin wo

M = 0.3

par fly karta hai. Kya area rule uski help karta hai?

Nahi.

M = 0.3

par koi shock nahi aur koi wave drag nahi, isliye

A (x)

smooth karne se kuch nahi milta; rule ek transonic/supersonic tool hai aur low Mach par idle rehta hai.

Edge case: Do bodies ka

A (x)

same hai lekin ek closed body of revolution hai aur doosra flat delta-wing planform. Same wave drag?

Haan, slender-body theory mein first order tak — identical

A (x)

identical wave drag force karta hai, aur yahi equivalence Whitcomb ne exploit ki thi (though higher-order aur roll-averaging effects unhe high

M

par alag karte hain).

Edge case: Agar fixed length mein volume

V \to 0

ho jaaye Sears–Haack formula mein, toh

D_{w}

kya karta hai, aur kya ye physical hai?

D_{w} \propto V^{2} \to 0

, isliye vanishingly thin body ka wave drag bhi vanish ho jaata hai — consistent, lekin aircraft ke roop mein useless, isliye real optimization

V

fixed rakhke length badhata hai.

Flashcards

Area rule total area par depend karta hai ya component areas par?

Sirf total

A (x)

par; fuselage, wings, tail ke beech ka split wave drag ko dikhta nahi.

Kink in

A (x)

A^{''}

mein kya produce karta hai?

Ek Dirac impulse (infinite spike), jise drag integral heavily penalize karta hai.

Kya area ruling friction aur induced drag katta hai?

Nahi — sirf wave drag, transonic/supersonic regime mein.

M \to 1

par Mach plane tilt?

μ = arcsin (1/ M) \to 9 0^{\circ}

, isliye Mach plane perpendicular ho jaata hai, Whitcomb's rule recover karta hai.