Mach number M = V - a — subsonic ( - 1), transonic (~1), supersonic ( - 1), hypersonic ( - 5): Question bank

3.1.4 · D5 · Physics › Compressible Flow & Aerodynamics › Mach number M = V - a — subsonic ( - 1), transonic (~1), sup

Ye traps char confusions ke aas-paas cluster hote hain: (1) Mach ko ek speed ki jagah ek ratio samajhna, (2) free-stream Mach ko local Mach se confuse karna, (3) ye bhool jaana ki sound adiabatic hoti hai, aur (4) boundaries ( $M = 0$ , $M = 1$ , $M \to \infty$ ) ko galat handle karna.

True ya false — justify karo

Kya "Mach 1" har jagah metres per second mein same number hota hai?

False.

a = γ R T

temperature par depend karta hai, isliye Mach 1 sea level par ~340 m/s hota hai lekin thandi stratosphere mein sirf ~295 m/s — Mach ek ratio hai

V / a

, koi fixed speed nahi.

Ek plane jo constant true airspeed par fly kar raha hai, uska Mach number hamesha constant hoga.

False. Agar wo thande air mein climb kare, toh

a

drop ho jaata hai, isliye wahi

V

ek zyaada

M

deta hai — plane altitude par "zyaada compressible" ho jaata hai bhaale speedometer same hi padhta ho.

Agar poora aircraft subsonic hai (

M < 1

), toh uske aas-paas hawa ka har parcel bhi subsonic hoga.

False. Hawa curved surfaces (wings) par accelerate hoti hai; free-stream

M \approx 0.85

par local flow

M = 1

se exceed kar sakta hai aur ek shock form kar sakta hai. Bilkul yahi cheez transonic flight ko mushkil banati hai.

Mach number dimensionless hota hai.

True. Ye speed divided by speed hai, isliye saari units cancel ho jaati hain — yahi reason hai ki ye compressibility ke liye gases aur scales mein universal knob hai.

Agar fixed temperature par pressure badhaao toh gas mein sound speed badh jaati hai.

False. Ideal gas ke liye

a = γ R T

sirf

T

par depend karta hai. Fixed

T

par

p

badhana

ρ

ko proportionally badhata hai, isliye

p / ρ

(aur hence

a

) unchanged rehta hai.

M = 1

par Mach cone flight path ke perpendicular ek flat wall ban jaata hai.

True.

sin μ = 1/ M = 1

μ = 9 0^{\circ}

milta hai, isliye "cone" completely flat khul jaata hai — saare emitted wavefronts ek single plane front mein pile up ho jaate hain.

Incompressible flow physics ka ek law hai jo

M = 0.3

ke neeche hold karta hai.

False. Ye ek modeling convenience hai:

M \approx 0.3

ke neeche fractional density change (

\sim \frac{1}{2} M^{2} \approx 4.5%

) itna chota hota hai ki ignore kar sako. Kuch switch off nahi hota — hum bas ek chote effect ko neglect karna choose karte hain.

Newton ka isothermal

a = p / ρ

sound ki sahi speed deta hai.

False. Sound itni fast oscillate karti hai ki heat leak nahi ho sakti, isliye ye adiabatic hai, jo

d p / d ρ = γ p / ρ

deta hai. Isse ek factor

γ \approx 1.18

add hota hai — Laplace ka correction jo experiment se match karta hai.

Error pakdo

"Kyunki jet Mach 2 par fly karta hai, aur sound 343 m/s hai, toh wo 686 m/s par fly karta hai."

Error sea-level

a

use karna hai. Cruise altitude par

T

bahut kam hota hai, isliye

a \approx 295

m/s aur

M = 2

matlab ~590 m/s hai. Tumhe local speed of sound use karni chahiye.

"Plane ne Mach 1 tod diya jis waqt usne 340 m/s exceed kiya."

Threshold

V = a

hai local temperature par, koi fixed 340 m/s nahi. Thandi high-altitude air mein

a

~295 m/s ho sakta hai, isliye Mach 1 ek lower true airspeed par cross hota hai.

"Density change

\frac{d ρ}{ρ} \sim M

ki tarah scale karti hai, isliye

M = 0.3

par hume 30% compression milega."

\frac{1}{2} M^{2}

ki tarah scale karta hai, linearly nahi.

M = 0.3

par wo

\frac{1}{2} (0.09) \approx 0.045

hai, yaani ~5%, yahi reason hai ki 0.3 practical cutoff hai.

M = 0.85

par sab kuch subsonic hai, isliye wing pressures ke liye main incompressible Bernoulli use karunga."

Do errors hain: wing par local flow supersonic ja sakta hai, aur

M = 0.85 ≫ 0.3

isliye density strongly vary karti hai — incompressible Bernoulli galat pressures deta hai. Isentropic compressible relations use karo.

"Mach angle

μ = arctan (1/ M)

hai."

sin μ = 1/ M

hai, jo

μ = arcsin (1/ M)

deta hai. Geometry wave radius

a t

ko hypotenuse

V t

(source travel) se compare karti hai, jo ek sine hai, tangent nahi.

"Wave-fixed frame mein hum accurate rehne ke liye

d ρ d a

term rakhte hain."

Ek weak (infinitesimal) sound wave ke liye,

d ρ d a

do tiny quantities ka product hai — second-order aur negligible. Isse rakhna us linearization ke against hai jo ek sound wave ko define karti hai.

"Kyunki

a^{2} = d p / d ρ

hai, aur pressure equals

ρR T

hai, toh

a^{2} = R T

hai."

Tum sirf

p = ρR T

T

constant maanke differentiate nahi kar sakte — process adiabatic hai, isliye

T

bhi

ρ

ke saath change hota hai. Sahi tarike se

d p / d ρ = γ p / ρ = γ R T

hai, jo extra

γ

deta hai.

Why questions

Kyun Mach number, raw speed ki jagah, compressible flow mein "sabse important single number" hai?

Kyunki ye decide karta hai ki fluid ke paas adjust karne ka time hai ya nahi: pressure signals

a

par travel karte hain, isliye

V / a

batata hai ki fluid warned hai (

M < 1

) ya overtake ho gaya (

M > 1

). Physics qualitatively ratio par change hoti hai, kisi bhi fixed speed par nahi.

Kyun sound ki speed

T

par depend karti hai, pressure par nahi?

Sound ek pressure pulse hai jo molecular collisions se pass hoti hai; hotter gas matlab faster molecules, isliye collisions pulse ko jaldi transmit karte hain. Zyaada

T

molecular speed ko

T

ki tarah badhata hai, aur

a

usse track karta hai.

Kyun disturbances "pile up" hokar shocks banate hain sirf tab jab

M > 1

hota hai?

Source apne emitted wavefronts ko overtake kar leta hai, isliye alag-alag times par emit kiye gaye waves overlap karte hain aur Mach cone envelope ke saath concentrate ho jaate hain, aage spread hone ki jagah — ek steep pressure jump, yaani ek shock.

Kyun transonic regime (

M \approx 1

) design karne ke liye sabse mushkil hai?

Subsonic aur supersonic pockets ek hi body par coexist karte hain, local shocks produce karte hain jo choti speed changes ke saath move karte hain. Mixed, shock-terminated flow unstable aur highly nonlinear hai — koi ek simple theory ise cover nahi karti.

Kyun hume

M \approx 5

se upar air ko simple perfect gas se zyaada treat karna padta hai?

Hypersonic speeds par shock layer itna hot ho jaata hai ki molecules vibrate, dissociate, aur ionize karte hain, energy absorb karte hain. Specific heats constant rehna band kar deti hain, isliye

a = γ R T

ke peeche calorically-perfect-gas assumption fail ho jaata hai.

Kyun Newton ka speed-of-sound prediction ~18% kam nikla?

Unhone assume kiya ki compressions isothermal hain (heat equalize karne ka time hai). Sound actually adiabatic hoti hai, jo gas ko ek factor

γ

se stiffer banata hai, aur

γ \approx 1.18

exactly woh missing ~18% hai.

Kyun wahi true airspeed high altitude par "zyaada compressible" feel hoti hai?

Thandi high-altitude air ka

a

lower hota hai, isliye wahi

V

ek higher

M

produce karta hai. Kyunki compressibility effects

M^{2}

ke saath scale karti hain, zyaada Mach matlab identical speed ke liye larger fractional density changes.

Kyun hum steady conservation laws use karke

a

derive kar sakte hain bhaale sound wave moving ho?

Hum wave-fixed frame mein jump karte hain taaki wave stationary ho aur gas steadily us mein se flow kare. Ye mushkil time-dependent equations ko simple steady mass aur momentum balances se swap kar deta hai.

Edge cases

Still air mein stationary object ka Mach number kya hoga?

M = V / a = 0/ a = 0

. Ye extreme incompressible limit hai — koi motion nahi, koi pressure disturbance nahi, koi compressibility ki chinta nahi.

Jab

M \to \infty

hota hai, toh Mach angle

μ

ka kya hota hai?

sin μ = 1/ M \to 0

, isliye

μ \to 0^{\circ}

— Mach cone ek needle-thin sliver mein collapse ho jaata hai jo flight path ke saath hug karta hai, matlab disturbances body ke aas-paas tightly wrapped rehti hain.

Kya Mach cone formula

sin μ = 1/ M

exactly

M = 1

par valid hai?

sin μ = 1 \Rightarrow μ = 9 0^{\circ}

deta hai, ek limiting flat front.

M < 1

ke liye iska koi solution nahi (

1/ M > 1

), jo sahi tarike se signal karta hai ki koi cone nahi banta jab source apni waves se slower ho.

Kya local Mach 1 exceed kar sakta hai jabki free-stream Mach 1 se kaafi neeche ho?

Haan — free-stream

M \approx 0.7

–

0.85

par ek wing par, acceleration local flow ko

M = 1

se past push kar sakta hai, ek shock create karta hai. Free-stream number ek value hai; flow field mein kai values hoti hain.

Exactly

M = 0.3

par, kya flow incompressible hai ya compressible?

Dono nahi — ye conventional cutoff hai jahan density change ~5% tak pahunchi hai. Ye ek soft, human-chosen boundary hai; physics smoothly vary karti hai, isliye 0.3 ko guideline maano, switch nahi.

M < 1

tumhe flow mein shocks ke baare mein kya batata hai?

Purely subsonic flow (

M < 1

har jagah) mein koi shocks nahi hote — har parcel pehle se warned hota hai aur smoothly adjust karta hai. Shocks ke liye kahin na kahin ek supersonic pocket chahiye, bhaale vehicle overall subsonic ho.

Wrap-up recall

Recall Traps ki one-line summary

Mach ek ratio hai $V / a$ , kabhi fixed speed nahi.
$a$ sirf temperature par depend karta hai (ideal gas ke liye), $T$ ke through.
Free-stream Mach ≠ local Mach; wings local flow ko faster banate hain.
Compressibility $\frac{1}{2} M^{2}$ ki tarah scale karti hai; $M = 0.3$ par ~5%.
Sound adiabatic hai → factor $γ$ , isothermal nahi.
$sin μ = 1/ M$ : $M = 1$ ke neeche koi cone nahi, $M \to \infty$ par needle-thin.

Related: Speed of Sound in Gases · Isentropic Flow Relations · Compressibility & Bernoulli's Limits · Prandtl–Glauert Correction · Reynolds Number