Boundary layer separation — adverse pressure gradient: Foundations

2.2.23 · D1 · Physics › Fluid Mechanics › Boundary layer separation — adverse pressure gradient

Parent note padhne se pehle, tumhe har wo letter apna banana hoga jo woh tumhare saamne phenk ta hai. Hum unhe ek ek karke milenge, har ek ke saath ek picture aur ek reason ki woh exist kyun karta hai.

Stage: yahan $x$ aur $y$ kya hain?

Is topic ki har cheez ek wall ke paas rehti hai. Humein pehle ek coordinate frame chahiye jo usi wall se chipka ho, room se nahi.

Figure 1 (neeche). Ek curved wall white mein jisme flow-direction axis $x$ (blue arrow) surface ko follow karte hue muda hua hai, aur height axis $y$ (yellow arrow) seedha perpendicular nikal raha hai. Neeche shaded region solid hai; wall line $y = 0$ hai. Notice karo kaise $x$ surface nahi chhodata chahe wall curve ho — is frame ka yahi to poora point hai.

Figure — Boundary layer separation — adverse pressure gradient

Velocity components $u$ aur $v$

Ek fluid particle ek direction mein move karta hai. Hum us motion ko do arrows mein split karte hain, ek har axis ke saath.

Streamline kya hota hai?

Word streamline Bernoulli section mein aane wala hai, toh ise abhi build karte hain.

Boundary layer, uski thickness $δ$ , aur edge

"Layer ke bahar" ke baare mein baat karne se pehle, humein precisely bolna hoga ki layer kahan khatam hoti hai. Us edge ka ek naam aur ek symbol hai.

Figure 2 (neeche). White wall $y = 0$ par, red dashed edge curve $y = δ (x)$ ke saath jo $x$ badhne par moti hoti jaati hai. Teen stations par, ek blue velocity profile dikhata hai $u$ wall par $0$ se shuru hoke red dot tak edge par free-stream value tak chadh ta hai; green arrows along-wall direction mark karte hain aur yellow arrows layer ke upar uniform free stream $U (x)$ dikhate hain. Red edge ko chadhte dekho — yahi $δ$ ka $x$ ke saath badhna hai.

Pressure layer ke across kyun change nahi karta ( $p = p (x)$ )

Parent note quietly assume karta hai ki $p$ sirf $x$ par depend karta hai, height $y$ par nahi. Yeh obvious nahi hai — aao actually ise derive karte hain.

Intuition Thin-layer scaling argument — piece by piece

Layer ko $x$ along length $L$ aur tiny height $δ$ do, jahan $δ ≪ L$ . Free-stream speed $U$ kaho. Do facts sab kuch drive karte hain:

Along-wall speed $u$ , $\sim L$ doori mein $\sim U$ range karta hai, toh $\partial u / \partial x \sim U / L$ .
Mass conservation, $\partial u / \partial x + \partial v / \partial y = 0$ , phir force karta hai $\partial v / \partial y \sim U / L$ bhi. Kyunki $v$ tiny height $δ$ mein change hota hai, iska matlab $v$ khud sirf $\sim U δ / L$ hai — chhota, kyunki $δ / L$ chhota hai.

Ab $y$ -equation mein har term ka wajan lagao:

Inertia $ρ u \partial v / \partial x \sim ρ U \cdot (U δ / L) / L = ρ U^{2} δ / L^{2}$ .
Compare karo along-wall pressure term se ( $x$ -equation se), jo order $ρ U^{2} / L$ hai.
Unka ratio hai $(ρ U^{2} δ / L^{2}) / (ρ U^{2} / L) = δ / L ≪ 1$ .

Toh har woh term jo pressure ko $y$ -direction mein bend kar sake, main flow terms se $δ / L$ factor se chhoti hai. Layer shrink karo ( $δ / L \to 0$ ) aur woh vanish ho jaate hain, sirf ye bachta hai: $\frac{\partial p}{\partial y} \approx 0.$

No-slip condition — wall par $u = 0$ kyun hota hai

Slope aur curvature: velocity profile padhna

$u$ ko height $y$ ke against sideways draw karo: woh curve velocity profile hai. Uski shape ke baare mein do cheezein sab kuch decide karti hain.

Figure 3 (neeche). Do velocity profiles ( $u$ horizontal, height $y$ vertical). Left (green): ek favourable profile — wall par steep slope aur sirf ek taraf bend, koi inflection nahi, healthy aur attached. Right (red): ek adverse profile — wall slope almost zero tak flat ho gayi hai aur ek yellow dot inflection point mark karta hai jahan bend direction switch karta hai. Wall slopes compare karo: green wala steep hai (strong forward pull); red wala almost flat hai (reverse hone wala hai).

Greek letters: $ρ$ , $μ$ , $ν$

Newton's law of friction: $τ_{w} = μ \partial u / \partial y$ kyun

Wall shear formula koi definition nahi hai jo aasman se giri ho — yeh fluid ke baare mein ek physical assumption par tikhi hai.

Pressure aur uska gradient

Ek bridge jo tumhare paas already hai: Bernoulli

Hum kehte rehte hain " $U (x) ↓$ forces karta hai $p (x) ↑$ ." Woh link Bernoulli's equation se aata hai — lekin sirf specific conditions mein, aur sirf ek specific form mein. Dono state karte hain. (Yaad karo upar se ki streamline ek aisi curve hai jis par fluid saath behta hai lekin kabhi cross nahi karta.)

Pieces topic ko kaise feed karte hain

Neeche diagram ek dependency map hai: har box is page ka ek idea hai, aur ek arrow " $A \to B$ " ka matlab hai $B$ ka koi matlab nahi jab tak $A$ apna na ho. Ise upar se neeche padho.

Equipment checklist

Daayein side cover karo aur parent note padhne se pehle har ek recite karo.

Is topic mein

x

aur

y

kya measure karte hain?

x

surface ke saath chalta hai (flow direction);

y

perpendicular, wall se door, wall

y = 0

par hai.

u

aur

U

mein kya farq hai?

u

layer ke andar actual along-wall speed hai (wall par

0

U

tak);

U (x)

fast free-stream speed hai edge

y = δ

par aur usse aage.

Streamline kya hota hai?

Ek curve jo har jagah velocity ke tangent ho; fluid ussi ke saath behta hai lekin use cross nahi karta, toh energy bookkeeping (Bernoulli) ek streamline ke saath ki jaati hai.

Boundary-layer thickness

δ (x)

kya hai?

Woh height jahan

u

U

99%

reach kar le; uske neeche friction matter karta hai, uske upar free stream hai, aur

δ

x

ke saath badhta hai.

Hum

p = p (x)

kyun likh sakte hain bina

y

-dependence ke?

y

-momentum equation mein har term main flow se

δ / L ≪ 1

factor se chhoti hai, toh

\partial p / \partial y \approx 0

; outer pressure seedha wall par stamp ho jaata hai.

No-slip condition kya kehti hai?

Stationary wall ko touch karne wala fluid frozen hota hai:

u = 0

aur

v = 0

y = 0

\partial u / \partial y

mein curly

\partial

kyun likhte hain

d

ki jagah?

u

dono

x

aur

y

par depend karta hai; partial ka matlab hai "sirf height

y

ke saath change,

x

fixed rakh kar."

\frac{\partial ^{2} u}{\partial y ^{2}}

physically kya represent karta hai?

Velocity profile ki curvature (bend) — uski slope kaise change hoti hai jab tum upar jaate ho; sign change inflection point deta hai.

μ

ρ

aur

ν

ko relate karo.

ν = μ / ρ

, matlab

μ = ρ ν

;

μ

stickiness hai,

ρ

density hai,

ν

momentum diffusivity hai.

Newton's law of friction aur uske peeche assumption state karo.

τ = μ \partial u / \partial y

, Newtonian fluid ke liye valid jahan shear stress linearly proportional hoti hai velocity slope se (air, water).

Wall shear stress likho aur batao ki woh separation kab signal karta hai.

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

; separation wahan shuru hoti hai jahan

τ_{w} = 0

, theek downstream mein backflow ke saath.

Pressure gradient "adverse" kab hota hai?

d p / d x > 0

— pressure downstream badhta hai, flow decelerate hoti hai (

d U / d x < 0

Bernoulli ki validity conditions aur uska differential form state karo.

Steady, incompressible, inviscid along a streamline; differentiate karne par milta hai

d p / d x = - ρ U d U / d x

, toh

U ↓\Rightarrow p ↑

Related builds: Boundary layer theory · Reynolds number & transition · Drag — form vs skin friction · Flow over a cylinder and sphere · Diffusers and nozzles · Stall on an aerofoil.