Boundary layer separation — adverse pressure gradient

2.2.23 · Physics › Fluid Mechanics

Separation hoti kyun hai aakhir?

Key yeh hai ki wall-layer fluid energy-poor hai lekin usse same pressure work karne ko kaha jaata hai jitna free stream karta hai.

Pressure gradient kya hai (aur "adverse" kyun)?

Naam kyun yeh hain: favourable gradient flow ko aage help karta hai; adverse wala usse oppose karta hai.

Separation condition kaise derive karein (first principles se)

Hum 2-D steady boundary-layer momentum equation use karte hain wall pe.

Step 1 — Thin layer ke andar x-momentum equation se shuru karein: $u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = - \frac{1}{ρ} \frac{d p}{d x} + ν \frac{\partial ^{2} u}{\partial y ^{2}}$ Yeh form kyun? Ek thin boundary layer mein pressure uske across vary nahi karta, isliye $p = p (x)$ outer flow dwara impose hota hai.

Step 2 — Wall pe, $y = 0$ pe evaluate karein. No-slip condition deta hai $u = 0$ aur $v = 0$ wall pe, isliye dono convective terms vanish ho jaate hain: $(u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y})_{y = 0} = 0$ Yeh step kyun? No-slip $u$ aur $v$ ko exactly wall pe kill kar deta hai, aur ek clean balance chhod jaata hai.

Step 3 — Jo bacha woh wall-curvature law hai. Bacha hua viscous balance $- \frac{1}{ρ} \frac{d p}{d x} + ν \frac{\partial ^{2} u}{\partial y ^{2}} = 0$ ko $ρ$ se multiply karein aur $μ = ρ ν$ use karein: $μ \frac{\partial ^{2} u}{\partial y ^{2}}_{y = 0} = \frac{d p}{d x}$ Yeh kyun important hai: wall pe velocity profile ki curvature sirf aur sirf pressure gradient se set hoti hai.

Step 4 — Separation precisely define karein. Separation wahan se shuru hoti hai jahan wall shear (aur isliye near-wall slope) pehli baar zero ho jaata hai:

Figure — Boundary layer separation — adverse pressure gradient

Worked examples

Common mistakes (Steel-manned)

Active recall

Recall Velocity profile ki wall curvature ke barabar kaunsi single quantity hoti hai?

$\frac{\partial ^{2} u}{\partial y ^{2}}_{w a l l} = \frac{1}{μ} \frac{d p}{d x}$ — pressure gradient ( $μ$ se divide karke).

Recall Separation point ki exact mathematical condition batao.

$τ_{w} = μ \partial u / \partial y ∣_{y = 0} = 0$ ; downstream near-wall slope negative ho jaata hai (backflow).

Recall Turbulent boundary layer laminar se baad mein kyun separate hoti hai?

Turbulent mixing high-momentum outer fluid ko wall tak laati hai, isliye woh adverse pressure gradient ko zyada der tak withstand karti hai.

Recall Ek 12-saal ke bachche ko explain karo (Feynman)

Socho tum cycle chal rahe ho aur road upar jaane lagti hai. Ek strong dost top pe aasani se nikal jaata hai; ek thaka hua baccha beech mein hi energy khatam kar leta hai, ruk jaata hai, aur peeche roll karta hai. Wing ya ball ke paas wali hawa woh "thaka hua baccha" hai — viscosity ne use slow kar diya tha. Jab pressure badhna shuru hota hai (upar ki taraf), woh slow hawa ruk jaati hai aur peeche roll karti hai, isliye smooth flow surface se chhil jaati hai aur ek messy swirling wake bana deti hai. Woh chhilna hi separation hai.

Flashcards

Adverse pressure gradient ko kya define karta hai?

d p / d x > 0

— pressure flow direction mein badhta hai, isliye outer flow decelerate hota hai (

d U / d x < 0

Wall pe

μ \partial^{2} u / \partial y^{2} ∣_{0}

kis ke barabar hota hai?

Streamwise pressure gradient

d p / d x

ke (momentum eqn se, jab no-slip convective terms ko kill kar deta hai).

Adverse gradient profile mein inflection point kyun force karta hai?

Wall curvature positive ho jaati hai (

d p / d x > 0

) jabki door bahar

u \to U

ke liye woh negative honi chahiye, isliye curvature sign change karti hai ⇒ inflection ⇒ instability/reversal.

Separation ki mathematical condition?

τ_{w} = μ \partial u / \partial y ∣_{y = 0} = 0

, just downstream backflow ke saath.

Diffusers (diverging ducts) easily kyun separate hote hain?

Area badhna ⇒

U

girta hai ⇒

p

badhta hai ⇒ adverse gradient.

Golf-ball dimples drag kyun reduce karti hain?

Woh layer ko turbulent trip karti hain; turbulent layers adverse gradients ko zyada der tak resist karti hain ⇒ separation delay hoti hai ⇒ narrow wake ⇒ kam form drag.

Kya separation high speed ki wajah se hoti hai?

Nahi — adverse pressure gradient ki wajah se hoti hai jo slow, momentum-poor near-wall fluid pe act karta hai.

Separation point ke thoda downstream kaun sa flow appear hota hai?

Reversed (recirculating backflow) wall ke paas, ek wake/separation bubble banaata hai.

Connections

Boundary layer theory — $δ$ , $δ^{*}$ , thin-layer assumption define karta hai.
Bernoulli's equation — layer edge pe $U (x)$ ko $p (x)$ se link karta hai.
Drag — form vs skin friction — separation pressure (form) drag govern karta hai.
Reynolds number & transition — laminar vs turbulent layer behaviour.
Flow over a cylinder and sphere — separation angle, vortex shedding.
Diffusers and nozzles — geometry se adverse vs favourable gradient.
Stall on an aerofoil — high angle of attack pe separation.