Boundary layer thickness, displacement thickness, momentum thickness: Question bank

2.2.21 · D5 · Physics › Fluid Mechanics › Boundary layer thickness, displacement thickness, momentum t

Poore page mein use hone wale symbols (sab parent mein build kiye hain):

$U$ — free-stream speed (wall se door fluid).
$u = u (y)$ — wall se $y$ height par local speed.
$δ$ — wo thickness jahan $u = 0.99 U$ .
$δ^{*}$ — displacement thickness (mass-flux deficit).
$θ$ — momentum thickness (momentum-flux deficit).
$H = δ^{*} / θ$ — shape factor.

True or false — justify karo

The boundary layer ka

y = δ

par ek sharp, physical edge hota hai.

False — velocity

U

tak sirf asymptotically approach karti hai, isliye

δ

ek convention hai (99% cutoff), fluid mein koi real wall nahi hai.

Kisi bhi non-trivial profile ke liye,

δ^{*}

δ

se chhota hota hai (aur generally

δ^{*} \leq δ

True — integrand

1 - u / U

strictly 1 se kam hota hai jahan bhi fluid kuch bhi move kar raha ho, isliye accumulated slab

δ

se thinner hota hai; equality

δ^{*} = δ

sirf degenerate case mein hoti hai jab

u = 0

for all

y < δ

(ek step profile), isliye universal statement

δ^{*} \leq δ

hai.

Order hamesha

θ < δ^{*} < δ

hota hai.

True kisi bhi physically reasonable (non-degenerate) profile ke liye —

θ

ke integrand mein extra factor

u / U < 1

use

δ^{*}

se neeche shrink kar deta hai, aur dono deficits hain isliye

δ

ke andar aate hain; sirf pathological step profiles strict inequalities ko collapse karti hain.

Free-stream speed

U

ko double karne se momentum thickness

θ

double ho jaati hai.

False —

θ

ek length hai jo dimensionless profile

u / U

se banti hai;

U

's cancel ho jaate hain, isliye ye profile shape aur

δ

par depend karta hai, directly

U

ki magnitude par nahi.

Bada shape factor

H

matlab boundary layer separation ke kareeb hai.

True — jaise profile wall ke paas flatter/inflected hoti hai (adverse pressure gradient),

δ^{*}

θ

ke relative badh jaata hai, isliye rising

H

Boundary layer separation ke liye ek early-warning flag hai.

Stationary wall ke liye,

u (0) = 0

ek assumption hai jo hum convenience ke liye lagate hain.

False — ye physical No-slip condition hai jo Viscosity enforce karti hai: surface par fluid molecules genuinely wall ke relative zero velocity rakhte hain.

δ^{*}

measure karta hai ki fluid downstream kitna displace hota hai.

False — ye measure karta hai ki outer inviscid streamlines kitni door outward (wall se door) push hoti hain missing mass flow ke liye jagah banane ke liye.

δ^{*}

formula ko

\infty

tak integrate karna hamesha exactly wahin number deta hai jitna

δ

tak integrate karne se milta hai.

Generally False — kyunki

u \to U

sirf asymptotically hota hai,

δ

ke aage ka tail ek chhota magar nonzero deficit area carry karta hai; dono integrals exactly tabhi agree karte hain jab hum explicitly assume karein ki

u (y) = U

for

y > δ

(standard modelling convention), warna us tiny tail se differ karte hain.

θ

directly wall par drag force se related hai.

True — momentum-integral (von Kármán) relation dikhata hai ki wall shear stress flat plate ke liye

ρ U^{2} \frac{d θ}{d x}

ke equal hai, isliye

θ

ki streamwise growth hi unit width per Skin friction drag hai (neeche "Why" section mein ek-line derivation dekho).

Error dhundho

"Momentum deficit

ρ (U - u)

hai, isliye

θ = \int (1 - u / U) d y

— same as

δ^{*}

Error ek factor

u

drop karna hai: momentum flux flowing mass (

ρ u

) times velocity gap

(U - u)

carry karta hai, jisse

ρ u (U - u)

milta hai, isliye

θ

mein extra

u / U

weight hai jo use

δ^{*}

se chhota banata hai.

δ^{*}

nikalne ke liye hum deficit ko

ρ U^{2}

se normalise karte hain."

Wrong normaliser —

δ^{*}

ek mass deficit hai, isliye hum ideal mass flux

ρ U

se divide karte hain,

ρ U^{2}

se nahi;

ρ U^{2}

θ

ke momentum balance ka hissa hai.

"Kyunki teeno thicknesses ki units length hain,

δ = δ^{*} + θ

Units match karna koi additive law imply nahi karta; teeno alag-alag conserved quantities (geometry, mass, momentum) measure karte hain aur aisa koi sum nahi hota — sirf inequality

θ < δ^{*} < δ

hai.

"Boundary layer us

x

par

δ = 5 x / R e_{x}

reach karne ke baad grow karna band kar deta hai."

Wo formula har

x

par

δ

hai; ye

x

ke saath badhta rehta hai kyunki

δ \propto x / U x / ν \propto x

— layer continuously downstream thickens hoti rehti hai.

"Sine profile ke liye

\int_{0}^{δ} sin (π y /2 δ) d y = δ

, isliye

δ^{*} = 0

Integral

2 δ / π \approx 0.64 δ

hai,

δ

nahi, jisse

δ^{*} = δ (1 - 2/ π) \approx 0.363 δ

milta hai — ek nonzero, positive deficit jaisa hona chahiye.

"Turbulent boundary layer ka

H \approx 3

hota hai, laminar se zyada, kyunki ye thicker hai."

Ulta hai — turbulent profiles fuller hoti hain (velocity wall ke paas steeply rise karti hai), isliye unka

H

laminar Blasius

H \approx 2.59

se lower (~1.3–1.4) hota hai; thickness

δ

shape se alag matter hai.

Why questions

Hume sirf

δ

ki jagah teen thicknesses ki zaroorat kyun hai?

Kyunki "layer kahan khatam hoti hai" ye fuzzy aur geometric hai; engineers ko un do conserved quantities ke precise measures chahiye jo wall steal karti hai — mass flux (

δ^{*}

) aur momentum flux (

θ

) — jo

δ

akela nahi de sakta.

θ

deficit ko

u / U

se weight kyun karta hai lekin

δ^{*}

nahi karta?

Mass flux sirf ek baar speed par depend karta hai (

ρ u

), isliye

δ^{*}

deficit ko ek baar dekhta hai; momentum flux do baar speed par depend karta hai (

ρ u \cdot u

), isliye

θ

mein actual mass-carrying speed

u / U

ka ek factor include karna padta hai.

θ

"directly drag se related" kyun hai — ek-line reason kya hai?

Plate par ek control volume lo: net momentum flux out total drag force ke equal hota hai (fluid ke liye Newton ka 2nd law). Wo momentum flux deficit exactly

ρ U^{2} θ

hai, isliye downstream iska rate of change,

τ_{w} = ρ U^{2} \frac{d θ}{d x}

, wall shear stress hai —

τ_{w}

ko plate par integrate karo aur tumhe Skin friction drag milta hai.

99% ka choice (95% ya 99.9% ki jagah)

δ^{*}

aur

θ

ke liye essentially harmless kyun hai?

Kyunki

y = δ

ke paas deficit integrands already almost zero ho jaate hain, isliye cutoff shift karne se integrals barely change hote hain —

δ^{*}

aur

θ

us convention ke liye

δ

se bahut kam sensitive hain.

Blasius solution

δ \propto x

kyun deta hai,

\propto x

nahi?

Wall se door momentum का diffusion (via Viscosity)

time

jaisa spread karta hai; plate ke paas fluid ka residence time

x / U

ke roop mein badhta hai, isliye diffused thickness

x / U \cdot ν \propto x

ke roop mein badhti hai.

Ek given

x

par zyada Reynolds number thinner boundary layer se kyun associated hai?

δ = 5 x / R e_{x}

: bada

R e_{x}

matlab inertia viscosity par dominate karta hai, isliye momentum wall se available time mein kam dur diffuse hoti hai, slowed region ko thin rakhte hue.

δ^{*}

ko streamlines outward push karne ke roop mein kyun interpret kar sakte hain?

Kyunki wall ke paas missing mass flow kahi na kahi jaani chahiye — outer inviscid flow wall se exactly

δ^{*}

thickness ke barabar door displace hoti hai, jo us missing mass ko carry karne wale ideal

U

-flow ki thickness hai.

Edge cases

y = 0

(wall par) par

δ^{*}

ka integrand kya hai?

1 - u / U = 1 - 0 = 1

ke equal hota hai, apna maximum value — deficit wall par sabse bada hota hai jahan fluid puri tarah stuck hai.

θ

ka integrand

\frac{u}{U} (1 - u / U)

y = 0

aur

y = δ

dono par kya hota hai?

Dono ends par zero —

y = 0

par factor

u / U = 0

hai, aur

y = δ

par factor

1 - u / U \approx 0

hai; momentum deficit layer ke beech mein kahi peak karta hai.

δ^{*}

\infty

tak integrate karna usually

δ

tak integrate karne ke same value kyun deta hai — aur kab nahi deta?

Modelling mein hum assume karte hain ki

u = U

for

y > δ

, tail integrand ko exactly zero banate hue; lekin true asymptotic profile ke liye

u \to U

sirf

y \to \infty

par hota hai, isliye ek tiny nonzero tail area exist karta hai — practice mein negligible, lekin literally zero nahi jab tak flat-tail assumption impose na ki jaye.

Ek idealised inviscid ("perfect") fluid ke liye jisme No-slip condition nahi hai,

δ

δ^{*}

, aur

θ

kya hain?

Sab zero — viscosity ke bina fluid freely slip karta hai,

u = U

har jagah, koi deficit nahi, aur koi boundary layer form hi nahi hoti.

Agar plate infinitely long ho, toh kya

δ

without bound badhti rehti hai?

Pure laminar formula mein ye

x

ke roop mein hamesha badhti hai, lekin physically layer critical

R e_{x}

ke baad turbulence mein transition karti hai, growth law change ho jaata hai — isliye laminar formula apply karna band ho jaata hai.

Boundary layer separation ke point par

θ

ka kya hota hai?

Wall shear stress zero ho jaata hai aur near-wall flow reverse ho jaati hai; simple deficit picture breakdown ho jaati hai kyunki

u

wall ke paas negative ho jaata hai, isliye profile-based integrals ek attached layer describe nahi karte.

Ek step profile ke liye (

u = 0

for

y < δ

, phir

u = U

δ^{*}

aur

θ

kya hain?

δ^{*} = δ

(

δ

tak pura deficit, isliye equality — degenerate case jahan

δ^{*} \leq δ

equals ban jaata hai) aur

θ = 0

(integrand

\frac{u}{U} (1 - \frac{u}{U})

wahan vanish hota hai jahan

u

0

U

ho) — ek unphysical limit jo dikhata hai ki

θ

ke liye ek gradual profile chahiye.

Kya

δ^{*}

kabhi negative ho sakta hai?

Sirf agar kahi

u > U

ho (fluid free-stream se faster, jaise wall jets ya strong favourable gradients mein); standard flat-plate boundary layer ke liye

u \leq U

, isliye

δ^{*} \geq 0