Boundary layer thickness, displacement thickness, momentum thickness

2.2.21 · Physics › Fluid Mechanics

Intuition Bada picture

Jab ek real (viscous) fluid kisi wall ke paas se flow karta hai, to fluid wall se chipak jaata hai — yahi no-slip condition hai. Surface par velocity bilkul zero hoti hai; door jaake yeh free-stream speed $U$ ke barabar ho jaati hai. Woh patli region jahan velocity $0$ se (almost) $U$ tak chadhti hai, woh boundary layer hai.

KYUN teen alag "thickness" ki zaroorat hai? Kyunki "boundary layer kahan khatam hoti hai?" yeh fuzzy sawaal hai (velocity asymptotically $U$ ke paas jaati hai). Engineers isliye teen precise, physically-meaningful measures define karte hain:

$δ$ — geometric thickness (jahan wall ka asar "mehsoos karna band" ho jaata hai)
$δ^{*}$ — wall ka flow deficit streamlines ko kitna bahar dhakelta hai
$θ$ — wall ki wajah se kitna momentum lost hota hai Har ek alag sawaal ka jawaab deta hai.

1. Boundary layer thickness $δ$

KYUN 99% aur 100% kyun nahi? Kyunki $u \to U$ sirf $y \to \infty$ par hota hai (asymptotic). Hume ek finite cutoff chahiye, isliye hum ek conventional threshold choose karte hain. 99% standard hai.

YEH KYA REPRESENT karta hai: slowed-down region ka ek geometric edge.

KAISE badhta hai: Flat plate par laminar flow ke liye (Blasius), $δ$ distance $x$ ke saath badhta hai: $δ \approx \frac{5 x}{R e _{x}}, R e_{x} = \frac{U x}{ν}$ To layer downstream mota hota jaata hai — wall jitni der kaam karti hai, utna zyada slowed fluid hota hai.

Figure — Boundary layer thickness, displacement thickness, momentum thickness

2. Displacement thickness $δ^{*}$ — derive karo

First principles se derivation (mass conservation):

Mass flux per unit width agar poora region $U$ par move karta (ideal): $\overset{m}{˙}_{i d e a l} = \int_{0}^{\infty} ρ U d y$

Actual mass flux (real, slowed): $\overset{m}{˙}_{r e a l} = \int_{0}^{\infty} ρ u d y$

Mass flux mein deficit: $Δ \overset{m}{˙} = \int_{0}^{\infty} ρ (U - u) d y$

Hum $δ^{*}$ ko ek ideal stream (speed $U$ ) ki thickness ke roop mein define karte hain jo yahi deficit carry kare: $ρ U δ^{*} = \int_{0}^{\infty} ρ (U - u) d y$

3. Momentum thickness $θ$ — derive karo

Derivation: Slow fluid ke dwara actually carry kiya gaya momentum hai $\int ρ u \cdot u d y$ . Woh momentum jo yahi same mass flow $ρ u$ carry karta agar $U$ par move karta, woh hai $\int ρ u \cdot U d y$ . Deficit: $Δ (momentum) = \int_{0}^{\infty} ρ u (U - u) d y$

Isko thickness $θ$ aur speed $U$ ki ek ideal slab ke momentum ke barabar rakho, yani $ρ U^{2} θ$ : $ρ U^{2} θ = \int_{0}^{\infty} ρ u (U - u) d y$

4. Worked Examples

5. Common Mistakes

6. Active Recall Flashcards

No-slip condition kya hai?

Ek solid wall par, fluid velocity wall ki velocity ke barabar hoti hai (stationary wall ke liye zero), isliye

u (0) = 0

Boundary layer thickness

δ

define karo.

Wall se woh distance jahan

u = 0.99 U

ho (free-stream speed ka 99%).

99% kyun aur 100% kyun nahi?

Kyunki

u \to U

sirf asymptotically hota hai; ek finite cutoff chahiye, 99% convention hai.

Displacement thickness ka formula.

δ^{*} = \int_{0}^{\infty} (1 - u / U) d y

δ^{*}

ka physical meaning.

Outer streamlines wall se kitni door push hoti hain = ek ideal

U

-stream ki thickness jo missing mass flux carry kare.

Momentum thickness ka formula.

θ = \int_{0}^{\infty} \frac{u}{U} (1 - u / U) d y

θ

ka physical meaning.

Ek ideal

U

-stream ki thickness jo wall ko lost hue momentum ke barabar momentum carry kare; wall drag se related.

θ

mein extra

u / U

factor kyun hai?

Momentum deficit

= ρ u (U - u)

: actual mass-flow

u

AUR velocity gap

(U - u)

dono chahiye.

Teeno thickness ka order.

θ < δ^{*} < δ

Shape factor

H

define karo aur uska laminar value batao.

H = δ^{*} / θ

; Blasius laminar

H \approx 2.59

; badhta

H

aane wale separation ka signal hai.

Linear profile

u / U = y / δ

ke liye

δ^{*}

θ

H

batao.

δ^{*} = δ /2

θ = δ /6

H = 3

δ

ke liye Blasius growth law.

δ \approx 5 x / R e_{x}

, jahan

R e_{x} = U x / ν

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tum apna haath paani mein flat karke chalate ho. Haath ko chhune waala paani uske saath khichta hai, lekin door ka paani kuch khaas parwah nahi karta. Haath ke paas ek patli "chipchipati layer" hoti hai jahan paani "stuck" se "free" ho jaata hai. Us layer ki height $δ$ hai. Ab, kyunki haath ke paas kuch paani slower move kar raha hai, kam paani slip karta hai jitna tum expect karte — aisa lagta hai jaise haath thoda mota hai aur paani ki ek patli slice block kar raha hai. Woh motaapan hai displacement thickness $δ^{*}$ . Aur haath paani se momentum (push) bhi chura leta hai, yahi drag banata hai — churi gayi push ki amount, thickness ke roop mein likhi, momentum thickness $θ$ hai. Teen rulers, teen alag cheezein measure karte hain.

Connections

No-slip condition — boundary layer ki physical origin.
Reynolds number — govern karta hai ki layer laminar hai ya turbulent.
Blasius solution — exact flat-plate laminar profile jo $δ = 5 x / R e_{x}$ deta hai.
Skin friction drag — momentum-integral equation ke through $θ$ se directly proportional.
Boundary layer separation — badhte shape factor $H$ se predict hota hai.
Viscosity — woh property jo wall par velocity gradient create karti hai.