Area rule — Whitcomb's rule for transonic drag reduction

3.1.26 · Physics › Compressible Flow & Aerodynamics

WHAT is the Area Rule?

$x$ = flight (streamwise) axis ke saath-saath distance.
$A (x)$ = body ka area jab ek plane flow ke perpendicular se kata jaaye ( $M \approx 1$ ke liye).
Supersonic $M > 1$ ke liye cutting plane Mach plane hoti hai, jo Mach angle $μ = arcsin (1/ M)$ par tilted hoti hai — yahi Supersonic Area Rule (Jones) hai.

WHY does only the area distribution matter?

Key physical chain:

Sudden change in A (x) \Rightarrow large \frac{d ^{2} A}{d x ^{2}} \Rightarrow strong flow acceleration \Rightarrow shock waves \Rightarrow wave drag.

HOW: Deriving the wave-drag formula from first principles

Hum linearized supersonic slender-body theory use karte hain. Ek slender body ka disturbance axis ke saath-saath strength $f (x)$ ke sources ki ek line se build kiya ja sakta hai. Jo local source strength ek body with cross-section $A (x)$ produce karta hai woh hai

f (x) = U_{\infty} \frac{d A}{d x},

Drag woh streamwise momentum hai jo waves mein kho jaata hai. Linearized supersonic Green's function (von Kármán, 1935) se carry through karne par, length $ℓ$ ke slender body ki wave drag von Kármán–Sears integral hai:

D_{w} = - \frac{ρ _{\infty} U _{\infty}^{2}}{4 π} \int_{0}^{ℓ} \int_{0}^{ℓ} A^{''} (x_{1}) A^{''} (x_{2}) ln ∣ x_{1} - x_{2} ∣ d x_{1} d x_{2}

jahan $A^{''} = \frac{d ^{2} A}{d x ^{2}}$ .

Slender bodies ke liye smooth curve kyun optimal hai: Sears–Haack body

Fixed length $ℓ$ aur fixed maximum area (ya fixed volume) ke saath sab $A (x)$ mein se $D_{w}$ minimize karna ek calculus-of-variations problem hai. Minimizer Sears–Haack body hai, jiski area distribution hai

A (x) = A_{m a x} [4 ξ (1 - ξ)]^{3/2}, ξ = \frac{x}{ℓ} \in [0, 1],

minimum wave drag (fixed volume $V$ ke liye) ke saath

D_{w} = \frac{9 π ^{2}}{2} \frac{ρ _{\infty} U _{\infty}^{2}}{ℓ ^{2}} (\frac{V}{ℓ})^{?} \sim \frac{128 q _{\infty} V ^{2}}{π ℓ ^{4}}

Figure — Area rule — Whitcomb's rule for transonic drag reduction

Worked Examples

Worked example Example 1 — "Coke bottle" fuselage kyun

Ek straight fuselage ek smooth $A_{fus} (x)$ deta hai. Mid-body par ek wing add karne se ek bump $A_{wing} (x)$ add hota hai jo wahin peak karta hai jahan wing sabse chodi hoti hai. Total $A_{tot} = A_{fus} + A_{wing}$ mein ab ek hump hai → wing par large $A^{''}$ → large wave drag.

Fix: fuselage ko exactly wahin narrow karo jahan wing area peak kare, taaki $A_{fus}$ dip kare aur wing ka hump cancel ho jaaye → $A_{tot}$ phir se smooth ho jaaye.

Yeh step kyun? Kyunki $D_{w}$ sirf $A_{tot} (x)$ par depend karta hai; uski curvature flat karne se directly drag-driving $A^{''}$ khatam hoti hai.

Result: Whitcomb ke wind-tunnel tests mein ~25–30% transonic drag reduction dikha; YF-102 Mach 1 exceed nahi kar sakta tha jab tak "area-ruled" karke F-102A nahi bana, jo phir kar sakta tha.

Common Mistakes (Steel-manned)

Flashcards

Transonic wave drag primarily kis quantity par depend karti hai, area rule ke according?

Total cross-sectional area

A (x)

ke longitudinal distribution par, chahe woh components mein kaise bhi banti ho.

Fuselage ko wings ke paas kyun pinch ("Coke-bottle") kiya jaata hai?

Taaki fuselage area dip kare aur wing ka area bump cancel ho, total

A (x)

smooth aur

A^{''}

small rakhe.

von Kármán–Sears integral mein wave drag kya drive karta hai?

Second derivative (curvature)

A^{''} (x)

— abrupt area changes/kinks se huge

A^{''}

milti hai → strong shocks → drag.

von Kármán–Sears wave-drag integral batao.

D_{w} = - \frac{ρ U ^{2}}{4 π} \iint A^{''} (x_{1}) A^{''} (x_{2}) ln ∣ x_{1} - x_{2} ∣ d x_{1} d x_{2}

Fixed length ke liye minimum-wave-drag area distribution kya hai?

Sears–Haack body:

A (x) = A_{m a x} [4 ξ (1 - ξ)]^{3/2}

ξ = x / ℓ

Fixed volume par Sears–Haack wave drag length ke saath kaisa scale karta hai?

D_{w} \propto V^{2} / ℓ^{4}

— length double karne se wave drag 1/16 ho jaati hai.

Supersonic (Jones) area rule mein

M \approx 1

se kya badalta hai?

Body ko

μ = arcsin (1/ M)

par tilted Mach planes se kaatnaa padta hai (roll par average), perpendicular planes nahi.

Kaunse real aircraft ne area rule prove kiya?

F-102A — reshape karke (area-ruled) usne Mach 1 toda, jo un-ruled YF-102 nahi kar sakta tha.

Kya area ruling skin-friction drag reduce karti hai?

Nahi — sirf Mach 1 ke paas aur upar wave drag; friction aur induced drag unaffected rehti hain.

Source strength

f (x) = U_{\infty} A^{'} (x)

kyun hai?

Ek growing cross-section ko rate

U_{\infty} d A

per length par volume displace karna padta hai, jo us strength ke line source ke barabar hai.

Recall Feynman: 12-saal ke bacche ko explain karo

Socho hawa ek plane ke paas se aise guzar rahi hai jaise paani ek smooth stick ke around slide karta hai. Agar stick beech mein achanak moti ho jaaye (wings ki wajah se), toh paani ko bahut tezi se side mein dhakka khana padta hai aur splash hota hai — sound ki speed ke paas woh splash ek shock wave hai jo plane ko peeche kheenchti hai. Whitcomb ne kaha: bas body ka middle wahin squeeze karo jahan wings bahar nikalti hain, taaki poori cheez phir se ek smooth, gently-fattening stick jaisi lagey. Hawa calmly slide karti hai, "splash" almost gayab ho jaata hai, aur plane same engine par aur tez jaata hai. Yahi pinched-waist shape ki wajah se kuch fast jets Coke bottle jaise dikhte hain!

Connections

Sears–Haack body — woh minimum-wave-drag shape jise area rule target karta hai.
Transonic flow — Mach ≈ 1 regime jahan shock-induced wave drag spike karta hai (transonic drag rise).
Shock waves and wave drag — physical loss mechanism jise area rule minimize karta hai.
Slender-body theory — von Kármán integral ke underlying source-distribution model.
Mach angle and Mach cone — supersonic area rule ke liye tilted cutting planes set karta hai.
Prandtl–Glauert and compressibility corrections — neighbouring compressible-flow tools.
Drag breakdown: friction, induced, wave — wave drag ko doosre drag types ke beech situate karta hai.