Oblique shock waves — θ-β-M relation: Question bank

3.1.13 · D5 · Physics › Compressible Flow & Aerodynamics › Oblique shock waves — θ-β-M relation

Definition Is page par use kiye gaye har symbol ka matlab

Sab parent note mein banaye gaye hain; yahan pura glossary hai taaki kuch bhi unexplained na lage.

$M_{1}$ = upstream Mach number (incoming flow speed ÷ local speed of sound).
$M_{2}$ = downstream Mach number — flow ka resultant Mach number shock cross karne ke baad, apni nayi (deflected) direction mein measure kiya gaya. Ye slowed normal part aur unchanged tangential part dono ko combine karta hai.
$β$ = shock angle (shock ka tilt incoming flow ke relative).
$θ$ = deflection angle (flow physically kitna shock ki taraf bend karta hai).
$w$ = tangential speed — velocity component ka woh hissa jo shock face ke saath-saath chalta hai. Poori velocity is piece mein plus ek normal piece (shock ke perpendicular) mein split hoti hai.
$M_{n 1} = M_{1} sin β$ = normal-component upstream Mach number (woh akela hissa jo normal shock ki tarah behave karta hai).
$M_{n 2}$ = shock ke baad normal-component Mach number (hamesha $< 1$ ).
$μ = arcsin (1/ M_{1})$ = Mach angle, sabse shallow possible wave.
$γ$ = ratio of specific heats (air ke liye $1.4$ ).

Figure — Oblique shock waves — θ-β-M relation

Picture: incoming flow $V_{1}$ shock (tilted magenta line) se angle $β$ par takraata hai. $V_{1}$ ko ek normal part $u_{n 1} = V_{1} sin β$ (seedha shock ke andar) aur ek tangential part $w = V_{1} cos β$ (uske saath-saath sliding) mein split karo. Sirf $u_{n 1}$ compress hota hai; $w$ bina kisi change ke pass ho jaata hai. Kyunki normal part shrink hota hai jabki $w$ constant rehta hai, resultant exactly $θ$ se neeche tip ho jaata hai.

True or false — justify

Ek oblique shock mein flow hamesha slow ho jaata hai.

True — normal component compress hota hai aur normal shock ki tarah slow hota hai, aur kyunki tangential part

w

unchanged rehta hai, total speed drop hoti hai;

M_{2} < M_{1}

hamesha hota hai, chahe

M_{2}

supersonic hi rahe.

Ek oblique shock downstream flow ko supersonic chhod sakta hai.

True — sirf normal component subsonic (

M_{n 2} < 1

) jaana zaroori hai; weak shock ke liye surviving tangential slide

w

resultant

M_{2} > 1

rakh sakta hai.

β = 9 0^{\circ}

par oblique shock zero deflection wala normal shock ban jaata hai.

True — relation mein

cot β = 0

, isliye

tan θ = 0

; poori velocity shock ke normal hai, kuch slide nahi hota, isliye koi direction nahi hai jismein bend ho sake aur flow seedha cross kar jaata hai.

Fixed wedge angle ke liye exactly ek hi possible shock angle hota hai.

False — θ-β curve ek hump hai, isliye har

θ < θ_{ma x}

ke liye do valid

β

hote hain: ek weak (chhota

β

) aur ek strong (bada

β

) solution.

Wedge angle badhane se shock angle hamesha badhta hai.

Weak branch par ek point ke baad false hai, aur

θ_{ma x}

ke baad bilkul false — jab maangi gayi turn

θ_{ma x}

se zyada ho jaaye to koi attached solution exist hi nahi karta aur shock detach ho jaata hai.

Tangential velocity conserved hoti hai kyunki flow "notice nahi karta" sideways shock ko.

Spirit mein True — shock face ke saath-saath koi pressure gradient nahi hota, isliye

w

par koi tangential force act nahi karta, hence tangential momentum (aur speed) unchanged rehti hai.

Mach wave bas vanishing strength ka oblique shock hai.

True — jab

θ \to 0

to numerator

M_{1}^{2} sin^{2} β - 1 \to 0

sin β = 1/ M_{1}

force hota hai, yaani

β = μ

Mach angle; "shock" koi jump nahi carry karta. Dekho Mach Angle and Mach Waves.

Oblique shock mein agar shock kaafi weak ho to entropy decrease ho sakti hai.

False — har compression shock entropy badhata hai; weak shocks ise bahut kam badhate hain (third-order small), lekin kabhi decrease nahi karte. Isliye expansion smoothly hota hai, Prandtl-Meyer Expansion ke zariye.

Spot the error

"Kyunki flow

M_{1} = 3

par shock cross karta hai, main

M_{1} = 3

normal-shock tables mein lookup karta hoon."

Galat Mach number — sirf normal component normal shock ki tarah cross karta hai, isliye tumhe

M_{n 1} = M_{1} sin β

use karna chahiye, jo chhota hai aur sahi (weaker) compression deta hai.

"Wedge flow ko turn karta hai, aur shock wedge surface ke saath baithta hai, isliye

β = θ

Nahi — shock surface se aage angle

β > θ

par khada hota hai; shock aur wall ke beech ka flow wall ke parallel chalta hai, lekin shock khud wall se zyada steeply inclined hota hai.

"Downstream Mach hai

M_{2} = M_{n 2}

, seedha normal-shock table se padh lo."

Geometry miss ho rahi hai —

M_{n 2}

sirf normal piece hai;

M_{2} = M_{n 2} / sin (β - θ)

se recover karo, kyunki downstream normal direction flow ke saath angle

(β - θ)

banata hai.

"Nature strong shock choose karta hai kyunki strong shock zyada stable hota hai."

External flow ke liye ulta hai — wedge ke liye open flight mein weak shock realize hota hai kyunki downstream pressure wahi free-stream ambient value hoti hai jo weak branch produce karta hai; strong branch ke liye ek higher imposed back-pressure chahiye hoti hai isse hold karne ke liye, isliye ye sirf confined/internal flows mein survive karta hai jahan ek wall ya throat woh pressure supply karta hai. "Stability" yahan koi inherent property nahi hai — jo bhi branch actual downstream pressure se match kare wahi hoga.

"Tangential velocity unchanged hai, isliye tangential Mach number bhi unchanged hai."

Bilkul sahi nahi — tangential speed

w

conserved hoti hai, lekin temperature (aur isliye sound speed

a

) shock ke across jump karti hai, isliye tangential Mach number

w / a

asal mein decrease karta hai.

"Agar

sin β < 1/ M_{1}

to hume bas ek bahut weak shock milta hai."

Koi shock nahi —

sin β < 1/ M_{1}

ka matlab hai

M_{n 1} < 1

, ek subsonic normal component, jo shock support nahi kar sakta; sirf

sin β \geq 1/ M_{1}

physical hai.

"Bow shock tab banta hai jab flow bahut fast hoti hai."

Speed nahi balki turning — bow shock tab appear hota hai jab maangi gayi deflection

θ

M_{1}

ke liye

θ_{ma x}

se zyada ho jaaye; ek blunt body ek aisi turn demand karta hai jo koi attached shock supply nahi kar sakta.

Why questions

Hum velocity ko normal aur tangential parts mein decompose kyun karte hain?

Kyunki shock physics (Rankine–Hugoniot jump) purely shock face ke across act karta hai; splitting se hum poori normal-shock machinery sirf normal component

u_{n 1}

par reuse kar sakte hain.

Tangential component

w

unchanged kyun survive karta hai jabki normal wala crush ho jaata hai?

Pressure sirf shock ke across jump karta hai, isliye force sirf normal direction mein act karta hai; sideways kuch push nahi kar raha, isliye sideways slide

w

bina kisi change ke continue karta hai.

M_{n 1} > 1

ek real oblique shock ke liye kyun zaroori hai, chahe

M_{1}

kitna bhi bada ho?

Shock ek supersonic-compression phenomenon hai apni crossing direction mein; agar normal component subsonic hota to shock karne ke liye koi supersonic flow nahi hoti, isliye wave Mach wave ya kuch nahi mein degenerate ho jaati hai.

θ-β curve mein maximum kyun hota hai baar baar badhte rehne ki jagah?

Numerator

M_{1}^{2} sin^{2} β - 1

ko prefactor

cot β

ke against trace karo: chhote

β

par numerator near zero hai isliye

θ \approx 0

;

β = 9 0^{\circ}

par

cot β

prefactor zero hai isliye

θ = 0

again. Kyunki

θ

zero se start hota hai, zero par end hota hai, aur beech mein positive hai, ise ek peak

θ_{ma x}

tak zaroor badhna aur phir girna hoga — wahi hump hai.

Weak shock "usually" kyun choose ki jaati hai lekin hamesha nahi?

Dono branches mathematically valid hain; kaun sa appear hoga ye downstream boundary conditions set karte hain — open flight pressure relieve karta hai aur weak choose karta hai, jabki ek required high back-pressure strong choose karta hai.

θ = 0

do alag situations se kyun correspond karta hai?

Relation ke

θ

ke do zeros hain — ek jahan numerator

M_{1}^{2} sin^{2} β - 1

vanish hota hai (

β = μ

, ek infinitely weak Mach wave) aur ek jahan

cot β

prefactor vanish hota hai (

β = 9 0^{\circ}

, ek full normal shock). Same zero deflection, opposite strengths.

Ek cone same angle ke wedge se pehle detach hone se pehle zyada bada flow Mach turn kyun tolerate kar sakta hai?

Three-dimensional relief — cone ke around flow azimuthally spread ho sakta hai, effective turn ko ease karta hai, isliye cone shocks un angles par attached rehti hain jo 2-D wedge ko detach kar dengi.

Edge cases

Fixed wedge angle ke liye

M_{1} \to \infty

par

β

ka kya hota hai?

β

ek finite minimum par approach karta hai, zero par nahi. Relation mein, jab

M_{1} \to \infty

M_{1}^{2}

terms top aur bottom mein dominate karte hain aur cancel ho jaate hain, bacha rehta hai

tan θ \to 2 cot β \frac{sin ^{2} β}{γ + cos 2 β}

— ek equation jisme koi

M_{1}

nahi bacha. Ek fixed

θ

isliye ek finite

β

fix karta hai; shock wedge ke aur kareeb ho jaati hai lekin kabhi flat nahi hoti uske upar, kyunki ek real deflection ke liye abhi bhi ek real normal component

u_{n 1}

chahiye.

Jab

M_{1} \to 1^{+}

(barely supersonic) to kya hota hai?

Mach angle

μ = arcsin (1/ M_{1}) \to 9 0^{\circ}

, isliye sabse weak wave bhi nearly normal khadi hoti hai, aur

θ_{ma x}

zero ki taraf shrink ho jaata hai — almost koi bhi turn shock detach kar deta hai.

Exactly

β = μ

par deflection kya hai?

Zero — ye Mach-wave limit hai jahan

M_{1}^{2} sin^{2} β - 1 = 0

, flow bina compression aur bina turn ke slide karta hai.

Agar

θ

thoda negative ho (expansion corner, convex turn) to kya?

Koi oblique shock nahi banta — compression shock flow ko sirf apne andar turn kar sakta hai; convex turn ko smoothly handle kiya jaata hai Prandtl–Meyer expansion fan ke zariye.

Exactly

θ = θ_{ma x}

par, shock weak hai ya strong?

Clearly koi bhi nahi — dono roots hump ke peak par ek mein merge ho jaate hain; thodi aur turning se koi attached solution nahi hota, isliye ye detachment se pehle last attached shock hai.

Agar

M_{n 1}

exactly

1

aaye to kaisa "shock" hai?

Zero-strength Mach wave —

M_{n 1} = 1

ka matlab hai koi compression jump nahi, isliye ye "genuine shock" (

M_{n 1} > 1

) aur "no shock" ke beech boundary mark karta hai.

θ_{ma x}

par theek

M_{2}

kya karta hai?

Wahan abhi bhi thoda supersonic hai. Curve ke saath

M_{2}

ko follow karo: jab

β

badhta hai shock strong hoti jaati hai aur

M_{2}

girta hai,

M_{2} = 1

(sonic) sirf peak ke baad reach hota hai, strong branch par thodi upar. Kyunki woh sonic point ek

β

par baitha hai jo

θ_{ma x}

dene wale

β

se thoda bada hai,

θ_{ma x}

par flow abhi bhi subsonic nahi — "

M_{2} = 1

" aur "

θ = θ_{ma x}

" points kareeb hain lekin identical nahi.

Recall Traps ka one-line summary

$M_{1}$ nahi balki $M_{n 1}$ use karo; tangential speed $w$ conserved hai lekin uska Mach number nahi; do shock roots (weak default); real shock ke liye $M_{n 1} > 1$ chahiye; aur $θ_{ma x}$ ke baad shock detach hokar bow shock ban jaata hai.