Prandtl-Meyer function ν(M): Question bank — Hinglish deep dive

3.1.17 · D5 · Physics › Compressible Flow & Aerodynamics › Prandtl-Meyer function ν(M)

Is page par use hone wale symbols aur conventions

Traps se pehle, har symbol ko pin down karte hain taaki koi item kisi undefined cheez par depend na kare. Ye list padhte waqt neeche diya figure dekho — usme corner, fan, aur velocity triangle dikhaya gaya hai jahan ye saare symbols rehte hain.

Definition Is page par har symbol

$M$ — Mach number: flow speed ko local speed of sound se divide karo. $M > 1$ ka matlab supersonic hai (sound se tez). Poora topic needs $M \geq 1$ .
$V$ — flow speed khud (metres per second), gas ki actual velocity magnitude. Picture mein ye lamba arrow hai; $M = V / a$ jahan $a$ local speed of sound hai.
$a$ — local speed of sound, ek tiny pressure ripple gas mein kitni tez travel karti hai. Ye gas ke thande hone par shrink hoti hai.
$γ$ — ratio of specific heats (gas ki ek fixed property: $γ \approx 1.4$ for air). Ye measure karta hai ki gas compress hone par kitni "springy" hai; ye ek pure number hai bina units ke aur $ν_{m a x}$ jaise numbers set karta hai.
$μ$ — Mach angle, $μ = arcsin (1/ M)$ : wo angle jo ek single Mach wave local flow direction se banata hai (figure mein thin blue line).
$θ$ — wall/flow ka turning angle.
$ν (M)$ — Prandtl–Meyer function, wo total angle jo ek sonic flow ne $M$ tak pahunchne mein "spend" kiya hai expand hote hue, jahan $ν (1) = 0$ .
$p, T$ — static pressure aur temperature (jo ek thermometer gas ke saath travel karte hue read karta hai).
$p_{0}, T_{0}$ — stagnation (total) pressure aur temperature: wo values jo gas tab reach karti agar use smoothly rest par laya jata. Isentropic flow mein $p_{0}$ aur $T_{0}$ constant rehte hain chahe $p$ aur $T$ change hon; ek shock $p_{0}$ ko lower karta hai.

True or false — justify karo

Ek flow sirf tabhi

ν (M)

use kar sakta hai jab wo supersonic ho.

True — poora construction

M = 1

par start hota hai aur Mach angle

μ = arcsin (1/ M)

sirf

M \geq 1

ke liye exist karta hai; ek subsonic flow mein koi Mach waves nahi hote jinse fan banana ho.

Prandtl–Meyer function sirf Mach number par depend karta hai, actual velocity ya temperature par nahi.

True —

ν (M)

sirf

M

(aur

γ

) ka ek pure function hai; derivation ne har

d V / V

d M / M

mein convert kar diya, toh saare dimensional quantities cancel ho gaye.

Flow ko

θ

se turn karna hamesha uska Mach number badhata hai.

False — sirf ek expansion (

θ > 0

, convex, door se turn hona)

M

badhata hai; ek compression turn use kam kar deta, lekin ek sharp compression ek shock banata hai aur

ν

tab apply nahi hota.

Static pressure aur static temperature dono expansion fan ke across drop hote hain.

True — flow isentropically accelerate karta hai, toh

p

aur

T

dono girte hain jabki stagnation quantities

p_{0}, T_{0}

constant rehte hain.

Stagnation pressure

p_{0}

ek Prandtl–Meyer fan ke across conserve hota hai lekin ek shock ke across nahi.

True — fan isentropic hai toh entropy (aur isliye

p_{0}

) unchanged rehta hai; ek shock entropy badhata hai,

p_{0}

girata hai.

ν (M)

bina bound ke badhta rehta hai jab

M \to \infty

False — ye finite ceiling

ν_{m a x} \approx 130.4 5^{\circ}

par approach karta hai (

γ = 1.4

ke liye);

arctan

terms

π /2

par saturate ho jaate hain.

Ek fixed turn angle

θ

ke liye, wohi

θ

incoming Mach number ki parwah kiye bina same downstream Mach number produce karta hai.

False —

θ = ν (M_{2}) - ν (M_{1})

ν

mein change fix karta hai, lekin kyunki

ν (M)

nonlinear hai, resulting

M_{2}

strongly depend karta hai is baat par ki

M_{1}

kahan hai.

Agar ek corner

ν_{m a x} - ν (M_{1})

se bada turn demand kare, toh flow simply

M = \infty

reach kar leta hai aur turning rok deta hai.

False — ideal inviscid limit

p \to 0

aur ek vacuum void predict karta hai, lekin real flow mein boundary layer pehle separate ho jaati hai (ya flow koi aur structure find kar leta hai); wall turning jaari rehti hai lekin flow physically utna door follow nahi kar sakta.

Prandtl–Meyer relation

θ = ν (M_{2}) - ν (M_{1})

ke liye zaroorat hai ki tum fan ki angular width

μ

jaano.

False —

θ

purely

ν

difference se aata hai; Mach angles

μ_{1}, μ_{2}

fan ki shape (front aur back edges) describe karte hain lekin

M_{2}

find karne ke liye inki zaroorat nahi.

Error dhundo

"Kyunki compression sirf negative expansion hai, main

θ = ν (M_{2}) - ν (M_{1})

use karunga

M_{2} < M_{1}

ke saath ek sharp concave corner ke liye."

Error: ek sharp compression corner ek shock produce karta hai, jo non-isentropic hai, aur

ν (M)

derive kiya gaya tha isentropic flow assume karke — tumhe oblique-shock relations use karne honge instead.

"Downstream Mach number pane ke liye, turn angle directly Mach number mein jodo:

M_{2} = M_{1} + θ

Error: tum

θ

ν

mein jodo,

M

mein nahi;

ν

aur

M

nonlinearly related hain, toh tumhe

M_{1} \to ν_{1} \to ν_{2} = ν_{1} + θ \to M_{2}

jaana hoga.

"Maine

ν (2) = 26.38

degrees mein lookup kiya aur

θ = 0.26

radians ka turn add karke

ν_{2} = 26.64

paya."

Error: units mix ho gayi —

26.38

degrees hai aur

0.26

radians hai; add karne se pehle ek common unit mein convert karo (

0.26 rad \approx 14. 9^{\circ}

), house rule ke according.

"Expansion fan flow mein ek discontinuity hai, bas ek smooth wali."

Error: ek fan bilkul bhi discontinuity nahi hai — flow properties uske through continuously vary karti hain; wohi continuity exactly reason hai ki ye isentropic rehta hai (shock discontinuity hai).

"Kyunki flow speed up hoti hai, density fan ke across badhni chahiye."

Error: expansion gas ko rarefy karta hai — density pressure aur temperature ke saath girti hai; speed badhti hai lekin fluid phail jaata hai.

"Fan mein Mach waves sab wall se same angle banate hain."

Error: har wave Mach angle

μ = arcsin (1/ M)

local flow se banata hai, aur kyunki

M

fan ke through badhta hai,

μ

shrink hota hai — waves alag alag angles par fan out hoti hain.

ν (1) = 0

ka matlab hai sonic flow ne zero turning ki hai, toh wo turn kar hi nahi sakti."

Error:

ν (1) = 0

sirf chosen zero reference hai; ek sonic flow phir bhi expand kar sakti hai — wo simply apna "turning budget" zero se shuru karti hai aur upar climb karti hai.

Why questions

Differential relation

d θ = \frac{M ^{2} - 1}{V} d V

mein

M

ki jagah

M^{2} - 1

factor kyun hai?

cot μ = M^{2} - 1

hai jo ek single Mach wave ke across velocity triangle (figure mein) se aata hai — velocity ko

μ

par inclined wave ke along aur across resolve karne par exactly ye cotangent milta hai.

d V / V

d M / M

mein convert karte waqt denominator

1 + \frac{γ - 1}{2} M^{2}

kyun appear karta hai?

Kyunki

V = M a

M

par directly aur local sound speed

a

ke through bhi depend karta hai; accelerating flow cool hoti hai, toh

a

girta hai aur

M

ke rise ko partly offset karta hai, wo damping denominator deta hai.

Ek expansion isentropic kyun rehti hai jabki ek compression nahi rehti?

Expansion mein Mach waves spread apart hoti hain aur kabhi cross nahi karti, toh koi discontinuity nahi banti; compression mein wo converge aur coalesce hokar ek shock ban jaati hain, jo entropy jump karta hai.

ν_{m a x}

finite kyun hai jabki

M

bina limit ke badh sakta hai?

Dono

arctan

terms

M \to \infty

par finite value

π /2

approach karti hain, toh unka weighted combination diverge karne ki jagah ek fixed ceiling par converge karta hai.

Hum ek differential relation ko

ν (M)

paane ke liye integrate kar sakte hain, lekin shock ke across kyun nahi?

Integration require karta hai ki process smooth aur reversible (isentropic) ho; ek shock ek finite discontinuity hai entropy jump ke saath, toh integrate karne ke liye koi smooth path nahi hai.

Tables

ν

ko degrees mein kyun dete hain jabki raw integral radians yield karta hai?

Geometric relation ka integral naturally ek angle radians mein produce karta hai; tables purely engineering convenience ke liye degrees mein convert karte hain — tumhe

θ

ko same unit mein rakhna hoga.

ν (M)

guaranteed invertible kyun hai (ek

ν

↔ ek

M

Iska integrand

\frac{M ^{2} - 1}{M ( 1 + \frac{γ - 1}{2} M ^{2} )}

saare

M > 1

ke liye positive hai, toh

ν

strictly increasing (monotonic) hai — har

ν

value ek unique

M

par map karti hai.

Edge cases

M = 1

par

ν

exactly kya hai, aur fan usse neeche kyun shuru nahi ho sakta?

ν (1) = 0

;

M = 1

ke neeche term

M^{2} - 1

imaginary ho jaata hai aur

μ = arcsin (1/ M)

undefined hai — expand hone ke liye koi Mach wave nahi hai.

M = 1

par Mach angle

μ = 9 0^{\circ}

hai — fan ke leading edge ke liye iska kya matlab hai?

Pehli (leading) Mach wave flow ke perpendicular khadi hai; jaise flow fan ke through accelerate karta hai, baad ki waves ever-shallower angles par aati hain.

Fan ki angular spread kya hogi agar incoming flow sirf thodi si supersonic ho, maan lo

M_{1} = 1.01

Leading wave almost normal hai (

μ \approx 8 2^{\circ}

) aur

ν \approx 0

, toh almost poora

ν_{m a x}

budget abhi bhi available hai — aisi flow bahut bade total angle se turn kar sakti hai.

Ek incoming flow ke liye jo already near

M_{1}

hai jahan

ν (M_{1})

ν_{m a x}

ke close hai, aur expansion ke baare mein kya special hai?

Bahut kam turning budget bachta hai, toh thoda sa extra

θ

bhi

M

ko infinity ki taraf aur

p

ko zero ki taraf drive karta hai — flow ideal vacuum limit ke kagar par hai (practically boundary layer pehle separate ho jaati hai).

Agar

θ = 0

ho (koi wall bend nahi), relation kya kehta hai?

ν (M_{2}) = ν (M_{1})

, toh

M_{2} = M_{1}

— koi turn nahi matlab koi expansion nahi aur Mach number unchanged hai, jaise expected.

Kya

ν (M)

kabhi ek real supersonic flow ke liye negative ho sakta hai?

Nahi — integrand

M > 1

ke liye

ν (1) = 0

se shuru hote hue saare jagah positive hai, toh

ν

hamesha

\geq 0

hai; ek "negative

ν

" ke liye

M < 1

chahiye, jo domain se bahar hai.

ν_{m a x}

physically kya determine karta hai, aur kya ye gas ke saath change hota hai?

γ

se set hota hai through

\frac{π}{2} (\frac{γ + 1}{γ - 1} - 1)

; ek alag gas (alag

γ

) ka alag ceiling hoga —

130.4 5^{\circ}

specifically

γ = 1.4

ke liye hai.

Kya hota hai physically agar ek corner flow se zyada turn demand kare jitna wo supply kar sake — vacuum, shock, ya separation?

Idealized inviscid answer hai ek vacuum void (

p \to 0

), lekin real viscous flow almost hamesha boundary layer par isse pahle separate ho jaati hai, aur confined geometries instead expansion ko ek shock mein terminate kar sakti hain — "pure vacuum" sirf frictionless limiting case hai.

Connections

Oblique Shock Waves — non-isentropic compression counterpart jinhe ye traps contrast karte rehte hain.
Mach Waves and Mach Angle — jahan $μ = arcsin (1/ M)$ aur $M = 1 \Rightarrow μ = 9 0^{\circ}$ edge case aata hai.
Isentropic Flow Relations — "pressure girta hai / density girta hai" items ke peeche $p / p_{0}$ , $T / T_{0}$ supply karta hai.
Expansion Fan / Centered Rarefaction — physical structure jiskī continuity True/False set probe karti hai.
Method of Characteristics — $ν (M) \pm θ$ invariants use karta hai, yahan establish ki gayi monotonicity par rely karke.
Nozzle Design (Supersonic) — in limits aur traps ka practical use.