Normal shock properties — M₂, P₂ - P₁, T₂ - T₁, ρ₂ - ρ₁, P₀₂ - P₀₁: Question bank

3.1.12 · D5 · Physics › Compressible Flow & Aerodynamics › Normal shock properties — M₂, P₂ - P₁, T₂ - T₁, ρ₂ - ρ₁, P₀₂

Shuru karne se pehle, ek shared vocabulary reminder taaki koi bhi symbol use hone se pehle define ho:

Definition Characters ki list

$M_{1}$ = Mach number just upstream (flow speed ÷ local sound speed shock se pehle); $M_{1} > 1$ hamesha.
$M_{2}$ = Mach number just downstream; $M_{2} < 1$ hamesha.
$u$ = gas ki flow speed (metres per second) — gas kitni tezi se shock ke through stream karti hai physically. $u_{1}$ upstream hai, $u_{2}$ downstream; gas slow down hoti hai, isliye $u_{2} < u_{1}$ .
Static properties $P, T, ρ$ = jo ek thermometer/barometer gas ke saath travel karte hue read karta hai. $P$ = pressure, $T$ = temperature, $ρ$ (rho) = density (mass per volume).
Stagnation properties $P_{0}, T_{0}$ = jo tab measure hota agar gas ko smoothly aur reversibly rest par laya jaata. $T_{0}$ total energy track karta hai; $P_{0}$ track karta hai ki us energy mein se kitni abhi bhi usable hai.
$γ$ (gamma) = ratio of specific heats, $γ = c_{p} / c_{v}$ . Ye gas ki ek fixed property hai jo batati hai ki wo kitni "springy" hai; ordinary air ke liye $γ = 1.4$ .
$c_{p}$ = specific heat at constant pressure — kitne joules lagte hain ek kilogram gas ka temperature ek degree badhane mein jab pressure fixed ho. ( $c_{v}$ wohi constant volume par hai; unka ratio $γ$ hai.)
$R$ = specific gas constant — ideal-gas law $P = ρR T$ mein per-kilogram constant. Ye neeche entropy formula mein bhi aata hai.
$Δ s$ = shock ke across entropy change; "irreversibility meter". Ek shock ke liye $Δ s > 0$ .

True or false — justify

True or false: Ek normal shock adiabatic hai, isliye wo isentropic bhi hai.

False. Adiabatic ka matlab hai koi heat boundary ke across nahi jaata, lekin entropy internally generate ho sakti hai violent, irreversible compression se; yahan

Δ s > 0

, isliye ye adiabatic hai lekin isentropic nahi.

True or false: Stagnation temperature

T_{0}

shock ke dono sides par same hoti hai.

True. Flow adiabatic hai aur koi shaft work nahi hoti, isliye total energy (aur isliye

T_{0}

) conserved hai — andar aata

c_{p} T_{0}

energy barabar hai bahar jaate

c_{p} T_{0}

energy ke.

True or false: Stagnation pressure

P_{0}

bhi conserved hai kyunki shock adiabatic hai.

False.

P_{0}

usable energy measure karta hai, jo entropy par depend karta hai via

P_{02} / P_{01} = e^{- Δ s / R}

(Eq. 11); kyunki

Δ s > 0

, stagnation pressure hamesha girta hai.

True or false: Kyunki energy conserved hai, shock ke across koi "loss" nahi hota.

False. Total energy conserved hai, lekin availability (work-extractable energy) lost hoti hai — entropy rise high-grade kinetic energy ko low-grade thermal energy mein degrade kar deta hai.

True or false: Kisi bhi

M_{1} > 1

ke liye downstream flow subsonic hoti hai.

True. Equation (7) deta hai

M_{2} < 1

har

M_{1} > 1

ke liye; second law reverse ko forbid karta hai (subsonic → supersonic across a shock).

True or false: Ek weaker shock (

M_{1}

1 ke karib) kam stagnation pressure lose karta hai.

True. Jab

M_{1} \to 1^{+}

, toh

Δ s \to 0

aur

P_{02} / P_{01} \to 1

(Eq. 11); limit mein shock ek lossless sound wave ban jaata hai.

True or false: Shock ko arbitrarily strong bana ke density ko boundless badhaya ja sakta hai.

False. Jab

M_{1} \to \infty

, Eq. (9) deta hai

ρ_{2} / ρ_{1} \to \frac{γ + 1}{γ - 1} = 6

air ke liye; mass conservation plus ek finite velocity drop compression ko cap kar deta hai, aur extra energy

T

aur

P

mein chali jaati hai.

True or false: Shock ke across static temperature aur static pressure dono badhte hain.

True. Gas compress aur decelerate hoti hai, isliye

P

T

, aur

ρ

sab upar jump karte hain jabki

u

aur

M

drop karte hain.

True or false: Velocity ratio

u_{1} / u_{2}

density ratio

ρ_{2} / ρ_{1}

ke barabar hota hai.

True. Continuity

ρ_{1} u_{1} = ρ_{2} u_{2}

directly rearrange hoti hai

u_{1} / u_{2} = ρ_{2} / ρ_{1}

mein (Eq. 9 mein middle equality); gas jis fraction se densify hoti hai, uski speed utni hi slow hoti hai.

True or false:

M_{1} = 1.5

ke teen shocks ek

M_{1} = 1.5

ke single shock jaitna total stagnation pressure lose karte hain.

False. Har shock apna khud ka entropy generate karta hai, isliye teen shocks ek se zyada

P_{0}

lose karte hain — lekin ek single strong shock se bahut kam jo poori Mach drop absorb kare, isliye staged oblique shocks use kiye jaate hain.

Spot the error

Galti dhundho: "Flow adiabatic hai, toh main

P_{2} / P_{1}

seedha isentropic relations se padhunga."

Isentropic relations assume karte hain

Δ s = 0

, lekin ek shock mein

Δ s > 0

hai; shock ke through normal-shock relations (8)–(11) use karo — isentropic relations sirf shock tak aur uske baad lagte hain, uspar nahi.

Galti dhundho: "

M_{1} = 0.8

, toh main equation (7) se shock properties compute karunga."

Koi shock nahi hai — shocks ke liye

M_{1} > 1

chahiye.

M_{1} < 1

ko (7) mein daalne se mathematically valid lekin physically forbidden

M_{2} > 1

milta hai jo entropy ko decrease karne ki maang karega.

Galti dhundho: "

T_{0}

conserved hai, aur

T

jump karta hai, isliye

M

badhna chahiye

T_{0} = T (1 + \frac{γ - 1}{2} M^{2})

balance rakhne ke liye."

Ulta hai: kyunki

T

badhta hai jabki

T_{0}

fixed hai, bracket chota hona chahiye, isliye

M

girna chahiye. Badhta static temperature exactly yahi reason hai ki

M_{2} < M_{1}

kyon hota hai.

Galti dhundho: "Ek stronger shock gas ko zyada heat karta hai, isliye downstream

T_{0}

zyada hoga."

T_{0}

shock strength se independent conserved rehta hai; jo ek stronger shock karta hai wo hai zyada kinetic energy ko static

T

mein convert karna (

T_{2}

badhata hai), jabki

T_{0}

incoming total energy par pinned rehta hai.

Galti dhundho: "Kyunki

P_{02} < P_{01}

, shock ne flow se energy remove kar di hai."

Koi energy remove nahi hui (

T_{0}

constant); shock ne order/availability remove ki hai. Same total energy, lekin usmein se kam ko useful work mein convert kiya ja sakta hai.

Galti dhundho: "Ek supersonic inlet mein main ek bada normal shock chahta hun flow ko efficiently slow karne ke liye."

Ek single strong normal shock bahut zyada stagnation pressure destroy kar deta hai (jaise

M_{1} = 5

par 94% lost); efficient inlets kaafi weak oblique shocks use karte hain bahut chhote total entropy rise ke saath decelerate karne ke liye.

Why questions

Kyon ek normal shock exist karta hai instead of ek smooth deceleration ke?

Downstream pressure disturbances sound ki speed par travel karti hain aur ek supersonic flow ko "warn" karne ke liye aage nahi nikal saktin; pressure pile up hota hai jab tak ek paper-thin discontinuity mein collapse nahi ho jaata — the shock.

Momentum equation

P + ρ u^{2}

kyon likha jaata hai sirf

P

ki jagah?

Kyunki control volume par net pressure force flow ka momentum flux change karta hai;

ρ u^{2}

wo momentum flux per area hai (density times speed times speed), isliye pressure aur momentum-flux milke shock ke across balance kiye jaate hain.

Density

\frac{γ + 1}{γ - 1}

par kyon saturate hoti hai lekin pressure aur temperature without bound badhte hain?

Continuity density ko velocity drop se tie karta hai, jo finite hai; jab velocity almost halt ho jaati hai,

ρ

aur nahi badh sakti, isliye shock ki remaining energy

P

aur

T

mein dump ho jaati hai.

Combined equations se hume do algebraic roots kyon milte hain, aur shock kaun sa hai?

Roots eliminate karne ke liye square karne se trivial root

M_{2} = M_{1}

introduce hota hai ("no shock"/sound-wave solution); genuine shock doosra root hai, (7), jo

Δ s > 0

satisfy karta hai.

Stagnation pressure "engineer's enemy" kyon hai jabki stagnation temperature nahi?

P_{0}

set karta hai ki engine kitna thrust/work extract kar sakta hai; ise lose karna directly performance cost karta hai.

T_{0}

energy balance se conserved hai, isliye wo "lost" nahi hoti fight karne ke liye.

Ek shock sirf supersonic ko subsonic mein hi kyon turn kar sakta hai aur kabhi reverse nahi?

Reverse process

Δ s < 0

carry karta, jo second law violate karta; entropy generation one-directional hai, isliye flow sirf

M_{1} > 1

M_{2} < 1

mein jump kar sakti hai.

Edge cases

Edge case: Jab

M_{1} \to 1^{+}

toh har property ratio ka kya hota hai?

Sab 1 ke paas aate hain aur

Δ s \to 0

— shock infinitesimally weak, isentropic sound wave mein degenerate ho jaata hai bina kisi jump ke.

Edge case: Air ke liye exact limit

M_{1} \to \infty

mein

M_{2}

kya hoga?

Eq. (7) se,

M_{2}^{2} \to \frac{γ - 1}{2 γ} = \frac{0.4}{2.8} \approx 0.143

, isliye

M_{2} \to 0.378

— downstream Mach number bottom out hoti hai, ye zero nahi jaata.

Edge case:

M_{1} \to \infty

ke saath

P_{02} / P_{01}

kya approach karta hai, aur physical meaning kya hai?

Ye 0 ki taraf tend karta hai — infinitely strong shock essentially saari usable stagnation pressure destroy kar deta hai, isliye hypersonic inlets catastrophic

P_{0}

loss suffer karte hain.

Edge case: Agar "shock" ka

M_{1} = 1

exact ho, kya ye abhi bhi shock hai?

Nahi —

M_{1} = 1

par koi jump nahi hota (

Δ s = 0

, sab ratios 1 ke barabar); ye boundary case sirf ek Mach wave hai, sabse weakest possible disturbance.

Edge case: Ek gas jisme

γ \to 1

(bahut saare internal degrees of freedom) — maximum density ratio ka kya hota hai?

\frac{γ + 1}{γ - 1} \to \infty

, isliye aisi gas ek strong shock se enormously compress ho sakti hai, kyunki energy internal modes mein hide ho jaati hai instead of density ki ceiling neeche push karne ke.

Recall Yahan har trap ki one-line summary

Do ideas jo almost har card resolve karte hain: (1) adiabatic ≠ isentropic — energy ( $T_{0}$ ) bachti hai, availability ( $P_{0}$ ) nahi; (2) second law arrow set karta hai — hamesha $M_{1} > 1 \to M_{2} < 1$ , kabhi reverse nahi, aur density compression capped hai jabki $P$ aur $T$ nahi.

Do chalkboard figures poore page ko visual banate hain: ek shock ka structure dikhata hai aur kaunsi quantities upar vs. neeche jump karti hain, doosra plot karta hai ki ratios $M_{1}$ ke saath kaise badhte hain (aur density kahan saturate hoti hai).

Figure — Normal shock properties — M₂, P₂ - P₁, T₂ - T₁, ρ₂ - ρ₁, P₀₂ - P₀₁