Foundations — Boundary layer thickness, displacement thickness, momentum thickness

2.2.21 · D1 · Physics › Fluid Mechanics › Boundary layer thickness, displacement thickness, momentum t

Yeh page assume karta hai ki aap kuch bhi nahin jaante. Pehle hum parent note the parent topic ka ek bhi formula padhne se pehle, usmein use hone wale har symbol ko samajhna zaroori hai. Upar se neeche padho — har block uske upar wale par depend karta hai.

1. Wall, fluid, aur unhe describe karne ke liye coordinates

Ek flat plate (wall) ko flat lete hue imagine karo, aur fluid — hawa ya paani — uske upar left se right ki taraf flow kar raha hai. Kisi bhi cheez ke baare mein baat karne ke liye humein do directions chahiye:

• $x$ = plate ke saath saath doori, leading edge (agla tip) se measure ki gayi. "Hum downstream kitni door hain?"
• $y$ = wall se seedha upar ki taraf doori, uske perpendicular. "Hum surface se kitna upar hain?"

Topic ko yeh kyun chahiye: har quantity — velocity, thickness — ek height $y$ par aur ek downstream position $x$ par measure ki jaati hai. Inhe do rulers ke bina hum keh nahin sakte ki kuch bhi kahan hai.

2. Speed: $u$, free-stream $U$, aur velocity profile ki picture

Do speeds baar baar aate hain. Inhe alag rakhna — ek capital $U$ hai, ek lowercase $u$ hai.

Kyunki $u$ height par depend karta hai, hum ise $u(y)$ likhte hain — padho "$u$ of $y$", matlab "height $y$ par jo speed milti hai". Agar aap $u$ ko horizontally aur $y$ ko vertically plot karo, toh ek curve milta hai jise velocity profile kehte hain.

Topic ko yeh kyun chahiye: teeno thicknesses is $u(y)$ curve ki shape se compute ki jaati hain. Har height par $u$ aur $U$ ke beech ka gap woh "deficit" hai jo wall ne create kiya hai.

3. No-slip condition — neeche wala arrow zero kyun hai

Yeh obvious nahin hai — aap soch sakte ho ki fluid freely slide karega. Lekin surface ke bilkul paas ke molecules uspe chip jaate hain. Poori kahani ke liye No-slip condition dekho.

Topic ko yeh kyun chahiye: no-slip hi wajah hai ki boundary layer exist karta hai. Iske bina, $u$ har jagah $U$ ke barabar hoga aur naapne ke liye kuch bhi nahin hoga.

4. Viscosity $\nu$ — woh "stickiness" jo slowing ko phailaati hai

Wall us fluid ko slow karti hai jo use touch kar rahi hai. Lekin slow hona upar ki layers tak kyun phailta hai jo wall ko touch nahin kar rahi? Kyunki fluid layers ek doosre par drag karti hain — yeh internal friction viscosity hai.

$\nu$ kahan se aata hai iske liye Viscosity dekho. Yeh bhi note karo $\rho$ (Greek "rho") = density, fluid ke har cubic metre mein packed mass (units $\text{kg}/{\text{m}}^3$). Humein $\rho$ chahiye kyunki parent note mass aur momentum track karta hai, aur dono is baat ke saath scale karte hain ki kitna material wahan hai.

Topic ko yeh kyun chahiye: viscosity woh messenger hai jo wall ka "slow down!" order baahri taraf, layer by layer, carry karta hai, §2 mein profile build karta hua.

5. Reynolds number $R{e}_x$ — flow smooth hai ya messy?

Wall ki slowing (viscosity) fluid ki aage badhne ki tendency (inertia) se ladti hai. Inke ratio se yeh decide hota hai ki boundary layer kaise behave karta hai.

Subscript $x$ yaad dilata hai ki yeh downstream jaane par badhta hai (bada $x$). Poore meaning ke liye Reynolds number dekho.

Topic ko yeh kyun chahiye: growth law $\delta \approx 5x/\sqrt{R{e}_x}$ (Blasius solution se, jahan $\delta$ §6 mein define hai) poori tarah $R{e}_x$ ke terms mein likha gaya hai. Bina yeh jaane ki $R{e}_x$ kya hai, aap woh formula nahin padh sakte.

6. Boundary layer thickness $\delta$ — layer "khatam" kahan hoti hai?

Hum baar baar $\delta$ mention karte rahe hain. Ab ise earn karne ka time hai. Problem yeh hai: $u$ $U$ ki taraf badhta hai lekin kabhi bilkul reach nahin karta — yeh $U$ tak pahunchta hai sirf jab $y\to \infty$ (asymptotically). Toh "boundary layer kahan rukti hai?" ka koi sharp jawab nahin hai. Engineers ek convention chun kar yeh fix karte hain.

Topic ko yeh kyun chahiye: $\delta$ layer ki height ka ruler hai. Yeh integrals (§8) ki upper limit set karta hai — $\delta$ ke baad deficit practically zero hai — aur Blasius growth law $\delta \approx 5x/\sqrt{R{e}_x}$ yahi predict karta hai.

7. Integral sign $\int$ — infinitely many patli layers ko jodna

Parent note thicknesses ${\int }_0^{\infty }(\cdots )dy$ jaise expressions se compute karta hai. Yeh beginners ko dara deta hai, lekin idea simple hai.

• ${\int }_0^{\infty }$ same idea hai $b$ ki jagah $\infty$ ke saath — "slices ko infinity tak oopar jab tak sum karte raho."
• Lekin boundary layer edge $\delta$ (§6) ke baad, $u\approx U$, isliye $(1-u/U)$ jaisa deficit $(1-1)=0$ ban jaata hai. Zeros add karne se kuch nahin badalta. Isi wajah se parent note kehta hai ki aap sum $\infty$ ki jagah $\delta$ par rok sakte ho — tail kuch contribute nahin karta.

Topic ko yeh kyun chahiye: ${\delta }^{\ast }$ aur $\theta$ defined hain integrals ke roop mein — yeh poori layer mein per-slice deficit ke totals hain.

8. Mass flux aur momentum flux — hum actually kya sum kar rahe hain

Do deficits (mass aur momentum) abstract lagte hain. Yahan height $dy$ aur width $1$ (per unit width) ki ek patli ribbon ki physical meaning hai:

Topic ko yeh kyun chahiye: ${\delta }^{\ast }$ ek mass-flux deficit measure karta hai (ek $u$), $\theta$ ek momentum-flux deficit measure karta hai (do $u$'s). $u$'s ki sahi sankhya paana sabse common galti ka akela sabse bada source hai.

9. Teeno thicknesses, ek hi nazar mein

Ab parent ke formulas mein har symbol earn ho chuka hai — teeno thicknesses side by side:

$\delta :u(y=\delta )=0.99U\text{(geometric edge, §6)}$

${\delta }^{\ast }={\int }_0^{\infty }(1-\frac{u}{U})dy,\theta ={\int }_0^{\infty }\frac{u}{U}(1-\frac{u}{U})dy$

• $\delta$ profile se padhi gayi ek height hai (99% cutoff), integral nahin — isliye isme $\int$ nahin hai.
• $1-u/U$ = height $y$ par fractional velocity gap (fluid wahan full speed se kitna kam chal raha hai). Yeh wall par $1$ hai, upar $0$ hai.
• $u/U$ = full speed ka fraction present — woh weight jo count karta hai ki "yahan actually kitna mass move kar raha hai."

Downstream, wall ka drag build up hota hai aur eventually flow ko poori tarah rook sakta hai — yahi Boundary layer separation hai, aur total wall friction jo yeh produce karta hai woh Skin friction drag hai. Dono $\theta$ se padhe jaate hain. (Parent note ${\delta }^{\ast }$ aur $\theta$ ko ek single ratio mein combine karta hai jise shape factor kehte hain — aap use wahan miloge; abhi aapko iska zaroorat nahin.)

Prerequisite map

Ise neeche se upar padho: coordinates aur speeds profile banate hain; no-slip (viscosity se) uski shape force karta hai; 99% cutoff $\delta$ deta hai; ribbons + flux humein deficits ko ${\delta }^{\ast }$ aur $\theta$ mein sum karne dete hain; Reynolds number $R{e}_x$ $\delta$ ko size karta hai.

Equipment checklist

Daayein taraf dhako aur khud ko test karo — aap parent note ke liye tabhi ready ho jab har line turant aa jaaye.