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

3.1.12 · Physics › Compressible Flow & Aerodynamics

Intuition Bada picture (WHY shocks exist karte hain)

Ek normal shock ek kaagaz-jitni patli region hoti hai (kuch mean-free-paths moti) jahan supersonic flow ko achanak subsonic speed tak slow karna padta hai. Kisi gas mein information sound ki speed se travel karti hai; jab flow sound se tez chalti hai, downstream disturbances incoming gas ko "raasta do" kehne ki warning nahi de sakti. Pressure pile up hota rehta hai jab tak ek discontinuity mein collapse nahi ho jaata — yahi shock hai. Iske paar jaate waqt, gas zordaar squeeze hoti hai: pressure, temperature, aur density upar jaate hain, velocity girta hai, aur — sabse important — entropy badhti hai (yeh process irreversible hai). Sirf ek cheez unchanged rehti hai — stagnation enthalpy (energy conserved rehti hai, flow adiabatic hai).

1. Teen conservation laws (first principles)

Shock ke aas-paas ek control volume lo. State 1 = upstream (supersonic), state 2 = downstream (subsonic). Steady, 1-D, adiabatic, koi area change nahi, CV walls par koi friction work nahi.

Ye teen kyun? Mass create nahi ho sakta, momentum sirf net force se change hota hai (yahan pressure difference), aur energy conserved hai kyunki koi heat add nahi hui aur shock koi shaft work nahi karta. Paanch unknowns ( $P_{2}, T_{2}, ρ_{2}, u_{2}, M_{2}$ ) hain, aur ye laws + equation of state system ko close karte hain.

2. Downstream Mach number $M_{2}$ derive karna (master result)

Sab kuch Mach number ke terms mein sabse saaf hota hai. $u = M a$ , $a^{2} = γ R T$ use karo, toh $ρ u^{2} = ρ γ R T M^{2} = γ P M^{2}$ .

Step A — momentum (2) rewrite karo: Kuch divide mat karo, bas $ρ u^{2} = γ P M^{2}$ substitute karo: $P_1(1+\gamma M_1^2) = P_2(1+\gamma M_2^2)\;\Rightarrow\; \frac{P_2}{P_1}=\frac{1+\gamma M_1^2}{1+\gamma M_2^2}\tag{4}$ Yeh step kyun? Momentum balance ko Mach numbers mein pure pressure ratio mein convert karta hai.

Step B — energy (3) rewrite karo: Stagnation temperature $T_{0} = T (1 + \frac{γ - 1}{2} M^{2})$ conserved hai, toh: $\frac{T_2}{T_1}=\frac{1+\frac{\gamma-1}{2}M_1^2}{1+\frac{\gamma-1}{2}M_2^2}\tag{5}$ Yeh step kyun? Energy conservation directly temperature ratio deta hai jab hum dono Mach numbers jaante hain.

Step C — mass (1) rewrite karo: $ρ u = \frac{P}{R T} M a = \frac{P M γ R T}{R T} = \frac{P M γ}{R T}$ . State 1 = state 2 set karo: $\frac{P_1 M_1}{\sqrt{T_1}}=\frac{P_2 M_2}{\sqrt{T_2}}\tag{6}$

Step D — combine karo. (4) aur (5) ko (6) mein daalo. Algebra ke baad (square karo taaki roots khatam ho jaayein) $P$ aur $T$ ratios collapse ho jaate hain ek single equation mein jo $M_{1}$ aur $M_{2}$ relate karta hai, jiska non-trivial root hai:

Yeh kyun beautiful hai: $M_{2}$ sirf aur sirf $M_{1}$ par depend karta hai. Trivial root $M_{2} = M_{1}$ "no shock" solution hai; equation (7) actual shock branch hai. Kisi bhi $M_{1} > 1$ ke liye, equation (7) $M_{2} < 1$ deta hai — supersonic hamesha subsonic ban jaata hai.

3. Property ratios (sab $M_{1}$ ke terms mein)

(7) ko wapas (4), (5) mein substitute karo, density ke liye mass use karo. Jab tak note na ho, $γ = 1.4$ maano.

4. Stagnation pressure loss $P_{02} / P_{01}$ — entropy ka fingerprint

$T_{0}$ conserved hai (adiabatic), lekin $P_{0}$ NAHI — yeh girta hai kyunki entropy badhti hai. Ideal gas ke entropy change $Δ s = c_{p} ln \frac{T _{2}}{T _{1}} - R ln \frac{P _{2}}{P _{1}}$ se, aur $P_{0} = P (1 + \frac{γ - 1}{2} M^{2})^{γ / (γ - 1)}$ se:

Kyunki $Δ s > 0$ sabhi $M_{1} > 1$ ke liye, $P_{02} / P_{01} < 1$ hamesha. Yeh loss engineer ka dushman hai: ek supersonic inlet mein, ek strong normal shock stagnation pressure ko barbad kar deta hai, engine thrust ko nuksan pahunchata hai. (Isliye kai weak oblique shocks prefer kiye jaate hain.)

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

5. Worked examples

6. Common mistakes (Steel-man + fix)

Recall Feynman: 12-saal ke bachche ko samjhao

Socho tum itna tez daudo ki hawa timely raasta nahi de sakti. Tumhare aage ek super-patli, super-squished hawa ki diwar ban jaati hai. Us diwar ko cross karte waqt, ek pal mein hawa hotter, bhaari, zyada pressure wali, aur sound se slower ho jaati hai. Energy save rehti hai (kuch jalaya nahi jaata), lekin squishing messy aur wasteful hai — tum ise kabhi perfectly un-squish nahi kar sakte. Woh "wasted neatness" hi reason hai ki hawa ki push-power (stagnation pressure) kyun girti hai, bhale hi uski total energy (stagnation temperature) same rehti hai.

Flashcards

Normal shock ke paar kya conserved hota hai?

Stagnation enthalpy / temperature

T_{0}

(adiabatic), saath mein mass, momentum, energy. Entropy NAHI, stagnation pressure NAHI.

Normal shock isentropic kyun nahi hota?

Yeh irreversible hai; entropy badhti hai (

Δ s > 0

), isliye isentropic relations iske paar use nahi ho sakte.

M_{1}

ke terms mein

M_{2}^{2}

ka formula

M_{2}^{2} = \frac{1 + \frac{γ - 1}{2} M _{1}^{2}}{γ M _{1}^{2} - \frac{γ - 1}{2}}

Shock ke paar pressure ratio

P_{2} / P_{1}

1 + \frac{2 γ}{γ + 1} (M_{1}^{2} - 1)

Density ratio

ρ_{2} / ρ_{1}

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

, equals

u_{1} / u_{2}

M_{1} \to \infty

par limiting density ratio (air)

\frac{γ + 1}{γ - 1} = 6

T_{2} / T_{1}

ko P aur ρ ratios se kaise nikaalte hain?

\frac{T _{2}}{T _{1}} = \frac{P _{2} / P _{1}}{ρ _{2} / ρ _{1}}

(ideal gas law).

P_{02} / P_{01} < 1

kyun hota hai?

Kyunki

Δ s > 0

hai, aur

P_{02} / P_{01} = e^{- Δ s / R}

M_{1} = 2

, air ke liye

M_{2}

aur

P_{2} / P_{1}

M_{2} = 0.577

P_{2} / P_{1} = 4.5

Kya normal shock subsonic flow mein ho sakta hai?

Nahi — iske liye

Δ s < 0

chahiye hoga, jo 2nd law violate karta hai.

M_{1} > 1

zaroori hai.

Connections

Isentropic Flow Relations — shock se pehle/baad valid, paar mein nahi.
Speed of Sound and Mach Number — kyun $M > 1$ trigger hai.
Oblique Shock Waves — turning + weaker $P_{0}$ loss; normal-shock physics se bana hai.
Rankine–Hugoniot Relations — in jumps ka $P$ – $ρ$ form.
Entropy and Second Law — stagnation-pressure loss ka source.
Supersonic Inlets and Diffusers — engineering reason ki hum normal shocks minimize kyun karte hain.
Stagnation Properties T0 and P0 — kya bachta hai vs kya lost hota hai.