Boundary layer separation — adverse pressure gradient: Visual walkthrough

2.2.23 · D2 · Physics › Fluid Mechanics › Boundary layer separation — adverse pressure gradient

Step 1 — Velocity profile kya hota hai? (woh picture jis par sab kuch tika hai)

Ek crucial baat: $u$ ek number nahi hai. Yeh $y$ ke saath badalta hai. Wall par fluid chipka rehta hai (friction se glued); door bahar yeh full free-stream speed $U$ par chalta hai. $u$ ka $y$ ke against graph velocity profile kehlata hai.

Figure s01 neeche pure page ka anchor hai: yeh woh picture hai jise hum baar baar redraw karte rahenge jab flow downstream march karta hai.

Figure — Boundary layer separation — adverse pressure gradient

s01 ko aise padho: horizontal axis speed $u$ hai, vertical axis wall ke upar height $y$ hai. Yellow curve follow karo — yeh ground ke paas jhuka hua hai (slow, chote blue arrows) aur upar zyada seedha hai (fast, lambe blue arrows), jab tak free stream $U$ tak nahi pahunchta. Origin par pink dot no-slip point hai. Woh jhuka hua shape hi puri kahani hai.

Step 2 — Profile ki slope hi wall friction hai

$y$ mein upar jaane par $u$ ki slope $\frac{\partial u}{\partial y}$ likhi jaati hai.

Ab hum wall par slope ki parwah kyun karte hain? Kyunki friction ek shearing cheez hai — fluid ki ek layer apne paas wali layer ko drag karti hai. Jitni tezi se speed ek gap mein badlati hai, utni hi tezi se layers rub karti hain. Woh rubbing force per unit area wall shear stress hai:

Figure s02 mein pink dashed line wall par profile ki tangent hai — iska steepness exactly $\partial u / \partial y ∣_{0}$ hai. Us dashed line ko dekho jab hum aage badhte hain: pure page ka pedagogical target woh moment hai jab yeh dashed tangent flat ho jaati hai (slope zero).

Sirf "speed" nahin balki $\partial u / \partial y$ tool kyun? Kyunki friction ko parwah nahi ki flow door mein kitni tezi se hai — use parwah hai ki speed tiny near-wall gap mein kitni jaldi badlati hai. Woh change ek slope hai, toh slope (derivative) exactly sahi tool hai. Yeh chota formula page ka hero hai: jab yeh slope zero ho jaaye, flow wall se haath chhudata hai. Hum baki page is baat ko prove karne mein lagaate hain.

Step 3 — Thin layer ke andar momentum equation

Hume ek aisa law chahiye jo puri layer mein $u$ ko govern kare, sirf wall par nahi. Woh law Newton ka second law hai ("mass × acceleration = force") jo boundary layer ke andar fluid particle ke liye likha gaya hai.

s03 mein yellow equation term-by-term colour-tagged hai: blue = acceleration (fluid kya kar raha hai), pink = pressure push, white = viscous smoothing. Yeh sirf $F = ma$ hai fluid ke kapdon mein.

Step 4 — Law ko wall par hi evaluate karo

Yahan hai clever move. $y = 0$ set karo — exactly surface par baitho — aur dekho do terms kaise khatam ho jaati hain. Lekin dhyan se noto: do alag boundary conditions khatam kar rahi hain, ek nahi.

Ab momentum equation ke left side ko dono conditions apply karke dekho:

$u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} ⟶ u = 0 (no-slip) 0 \cdot \frac{\partial u}{\partial x} + v = 0 (impermeability) 0 \cdot \frac{\partial u}{\partial y} = 0$

Pehla term $u$ se multiply hai, jo no-slip se $0$ hai → gone.
Doosra term $v$ se multiply hai, jo impermeability se $0$ hai → gone.

Toh wall par acceleration terms bilkul khatam ho jaati hain. Newton ka law do right-hand forces ke beech ek clean balance mein collapse ho jaata hai:

$- \frac{1}{ρ} \frac{d p}{d x} + ν \frac{\partial ^{2} u}{\partial y ^{2}}_{y = 0} = 0$

Figure s04 mein do vanishing terms ko strikethrough kiya gaya hai aur us condition se label kiya gaya hai jo har ek ko kill karti hai, toh tum dekh sako kyun left side collapse hota hai.

Yeh step pura trick kyun hai: layer mein baaki jagah messy convective terms humse ladte hain. Exactly wall par, do wall conditions unhe free mein erase kar deti hain, sirf pressure vs. viscosity bachti hai. Yeh khade rehne ki sabse clean jagah hai.

Step 5 — Wall-curvature law

Ab tidy up karo. Balance ko $ρ$ se multiply karo aur $μ = ρ ν$ use karo:

$- \frac{d p}{d x} + ρ ν \frac{\partial ^{2} u}{\partial y ^{2}}_{y = 0} = 0 ⟹ μ \frac{\partial ^{2} u}{\partial y ^{2}}_{y = 0} = \frac{d p}{d x}$

Yeh boxed result kuch beautiful kehta hai: pressure gradient akele decide karta hai ki velocity profile wall par kaise bend karega. Koi guessing nahi — $d p / d x$ ka sign hi wall curvature ka sign hai.

Figure s05 boxed law ko frame karta hai aur do cup/cap shapes dikhata hai jo yeh select karta hai — jab bhi $d p / d x$ ka sign aaye iske paas waapas aao.

Hum sirf slope ki nahin balki curvature ki parwah kyun karte hain: slope batata hai friction abhi kya hai; curvature batata hai profile fat hone wali hai ya hollow — trouble ki early warning.

Step 6 — Favourable gradient: profile fat aur healthy rehta hai

$\frac{d p}{d x} < 0$ lo — wall ke saath pressure girta hai (ek favourable gradient; opposite duct ke liye dekho Diffusers and nozzles).

Boxed law se, negative $d p / d x$ ka matlab hai wall curvature $\partial^{2} u / \partial y^{2} ∣_{0}$ negative hai: profile bilkul bottom se cap over (⌢) karta hai. Toh curve wall se tezi se upar jaata hai aur smoothly free stream $U$ mein round ho jaata hai — koi bending back nahi, koi inflection nahi.

Wall par steep → bada $\partial u / \partial y ∣_{0}$ → bada $τ_{w}$ → strong forward grip.
Near-wall fluid ko help along kiya ja raha hai; yeh rukne wala nahi.

Figure s06 mein, notice karo ki blue dashed tangent wall par steep hai, aur yellow curve upar tak fat hai — ise s07 mein aane wale sick profile ke saath side-by-side compare karo.

Yeh ek khush, attached boundary layer hai. Kuch bhi separate nahi hota.

Step 7 — Adverse gradient: profile mein ek inflection point force ho jaata hai

Ab villain: $\frac{d p}{d x} > 0$ — pressure badh raha hai (adverse gradient).

Boxed law ab wall curvature ko positive banata hai: wall ke paas profile door bend karta hai (⌣). Ab hume argue karna hoga ki upar kahin negative curvature ka region hai — yahan hai honest chain:

Wall par, curvature positive hai: $\frac{\partial ^{2} u}{\partial y ^{2}}_{0} = \frac{1}{μ} \frac{d p}{d x} > 0$ .
Door bahar, flow constant free stream par settle ho jaata hai, $u \to U$ . Flat line ki zero slope aur zero curvature hoti hai, toh $\frac{\partial ^{2} u}{\partial y ^{2}} \to 0$ jab $y$ bada hota hai.
Us flat top ke paas pahunchte waqt, slope $\partial u / \partial y$ positive hai (abhi bhi chadh rahi hai) lekin zero tak decrease hona chahiye. Ek positive quantity jo decrease ho rahi hai ka negative rate of change hota hai — aur slope ki rate of change hi curvature hai. Toh free stream ke bilkul neeche curvature negative hai.
Curvature isliye wall par positive aur free stream ke bilkul neeche negative hai. Ek continuous quantity jo sign change karti hai woh beech mein zero se guzarti hai — ek inflection point.

Inflection point ek known danger sign hai: S-shaped profile wall ke paas bahut slow, easily-reversed fluid ka ek pocket carry karta hai. Adverse pressure — us already-slow fluid par backward push karne wali force — wahi hai jo ise tip over karti hai.

Figure s07 teeno regions explicitly mark karta hai: neeche cup (⌣), blue inflection dot jahan curvature $= 0$ , aur upar cap (⌢) jahan curve $U$ mein round hota hai.

Yeh fuel-tank picture se kaise connect hota hai: energy-poor wall fluid (slow, viscosity se) ek backward push (rising pressure) se milta hai. Inflection point us mismatch ka mathematical fingerprint hai. Dekho Flow over a cylinder and sphere jahan yeh real body par bite karta hai, aur Stall on an aerofoil wing version ke liye.

Step 8 — Separation point: wall slope zero ho jaata hai

Flow ko steadily adverse gradient ke neeche downstream follow karo. Near-wall slope $\partial u / \partial y ∣_{0}$ shrink karta jaata hai (profile bottom par zyada se zyada upright lean karta hai). Exactly jis instant yeh zero ho jaata hai woh separation hai:

Is point ke bilkul downstream slope negative ho jaata hai: $\frac{\partial u}{\partial y}_{0} < 0$ , matlab wall par bilkul fluid ab backward creep kar raha hai — reversed (backflow) flow. Upar forward flow, neeche backflow → boundary layer surface se lift off hoti hai aur wake mein roll ho jaati hai. Yeh form drag ko feed karta hai.

Figure s08 mein yellow curve (separation: wall par flat tangent) ko pink dashed curve (bilkul downstream: profile wall ke paas $u < 0$ tak poke karta hai) se compare karo. Blue dot $τ_{w} = 0$ mark karta hai.

Ek-picture summary

Figure s09 char key profiles ko wall ke saath side by side rakhta hai — ise left to right ek movie ki tarah padho jab flow downstream travel karta hai.

Upar sab kuch wall ke saath profiles ki ek march mein compress ho jaata hai:

Favourable region ( $d p / d x < 0$ ): fat profile, wall par steep, koi inflection nahi → attached.
Adverse region ( $d p / d x > 0$ ): boxed law $μ \partial^{2} u / \partial y^{2} ∣_{0} = d p / d x$ positive wall curvature force karta hai, jabki curve ko $U$ mein flatten hona hai (upar negative curvature) → ek inflection point appear hota hai → profile base par thin ho jaata hai.
Separation point: wall slope zero ho jaata hai → $τ_{w} = 0$ (blue dot marked SEP).
Downstream: wall slope negative → backflow → layer wake mein peel off ho jaati hai.

Recall Pure walkthrough ki Feynman retelling

Velocity profile bas ek picture hai "wall ke upar har height par hawa kitni tezi se hai". Ground par hawa glued still hai (no-slip), upar yeh full speed move karti hai. Us picture ki slope ground par friction hai — hawa wall ko kitna hard grip karti hai. Humne thin layer ke andar ek fluid speck ke liye Newton ka law likha, phir exactly wall par khade ho gaye: kyunki hawa wall ke saath slide nahi kar sakti (no-slip → speed zero) aur solid wall se guzar nahi sakti (impermeability → cross-speed zero), do pure terms vanish ho jaate hain aur hum pressure aur viscosity ke beech ek tidy tug-of-war par reh jaate hain. Woh ek gem deta hai: profile wall par kaise bends karta hai yeh purely is baat se decide hota hai ki pressure badh raha hai ya gir raha hai. Girta hua pressure (downhill) → profile nice aur fat, hawa hard grip karti hai, sab attached. Badhta hua pressure (uphill) → wall ek taraf bend karta hai lekin top ko free stream mein doosri taraf flatten karna hoga, toh neeche ek S-bend easily-shoved hawa ke saath appear hota hai; aage chalte raho aur ground-level slope zero tak drop ho jaata hai — woh separation hai — aur just past isme near-wall hawa backward roll karti hai aur smooth flow peel away ho jaata hai.

Active recall

Recall Wall par convective terms ko kaun si do boundary conditions kill karti hain, aur har ek kya fix karta hai?

No-slip tangential speed fix karta hai ( $u = 0$ ); impermeability normal speed fix karta hai ( $v = 0$ ). Saath milkar yeh dono convective terms zero kar deti hain.

Recall Wall par profile ki curvature ka sign kya set karta hai?

$d p / d x$ ka sign, $μ \partial^{2} u / \partial y^{2} ∣_{0} = d p / d x$ ke zariye. Adverse ( $> 0$ ) → positive wall curvature.

Recall Adverse gradient kyun necessarily ek inflection point create karta hai?

Wall curvature positive hai; lekin jab $u \to U$ toh slope zero tak decrease hota hai, jo free stream ke bilkul neeche negative curvature force karta hai. Positive-to-negative matlab curvature zero se guzarti hai — ek inflection.

Recall Separation condition state karo aur bilkul downstream kya hota hai.

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