Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions of M

3.1.7 · Physics › Compressible Flow & Aerodynamics

Stagnation state kya hota hai?

KYA chahiye: teen ratios $\frac{T}{T _{0}}, \frac{P}{P _{0}}, \frac{ρ}{ρ _{0}}$ purely $M$ aur $γ$ ke terms mein.

First principles se derivation

Step 1 — Energy equation se $T / T_{0}$ milta hai

Steady adiabatic energy equation (per unit mass) ek perfect gas ke liye:

$h + \frac{1}{2} V^{2} = h_{0} = const$

Ye step kyu? Koi heat nahi, koi work nahi ⇒ stagnation enthalpy $h_{0}$ conserved hai. Stagnation point par $V = 0$ isliye $h_{0}$ wahan ki enthalpy hai.

Ek perfect gas ke liye $h = c_{p} T$ :

$c_{p} T + \frac{1}{2} V^{2} = c_{p} T_{0}$

$c_{p} T$ se divide karo:

$\frac{T _{0}}{T} = 1 + \frac{V ^{2}}{2 c _{p} T}$

Ab $V^{2} / (c_{p} T)$ ko $M$ ke terms mein express karna hai. In do facts ka use karo:

Speed of sound: $a^{2} = γ R T \Rightarrow V^{2} = M^{2} a^{2} = M^{2} γ R T$ .
$c_{p} = \frac{γ R}{γ - 1}$ ( $c_{p} - c_{v} = R$ aur $γ = c_{p} / c_{v}$ se).

Ye step kyu? Ye velocity ko Mach number mein convert karta hai, jo ki humara ek maatra variable hai.

$\frac{V ^{2}}{2 c _{p} T} = \frac{M ^{2} γ R T}{2 \cdot \frac{γ R}{γ - 1} T} = \frac{γ - 1}{2} M^{2}$

Step 2 — Isentropic relations se $P / P_{0}$ aur $ρ / ρ_{0}$ milta hai

Ek perfect gas ke isentropic process ke liye:

$\frac{P _{0}}{P} = (\frac{T _{0}}{T})^{\frac{γ}{γ - 1}}, \frac{ρ _{0}}{ρ} = (\frac{T _{0}}{T})^{\frac{1}{γ - 1}}$

Ye step kyu? $P v^{γ} =$ const aur $P v = R T$ se, $v$ eliminate karne par $P \propto T^{γ / (γ - 1)}$ aur $ρ \propto T^{1/ (γ - 1)}$ milta hai. Flow isentropic hai, isliye ye stagnation aur local states ko link karte hain.

$T_{0} / T$ result substitute karne par:

Consistency check (Feynman test): divide karne par $\frac{P _{0} / P}{ρ _{0} / ρ} = (1 + \frac{γ - 1}{2} M^{2})^{1} = \frac{T _{0}}{T}$ milta hai, jo exact ideal-gas law $P / ρ \propto T$ hai. ✓

Figure — Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions of M

"Tables" padhna

Table bas in teen formulas ko $γ = 1.4$ (air) ke liye bahut saare $M$ values par evaluate karna hai. Key memory anchors:

$M$	$T / T_{0}$	$P / P_{0}$	$ρ / ρ_{0}$
0	1.000	1.000	1.000
0.5	0.9524	0.8430	0.8852
1.0	0.8333	0.5283	0.6339
2.0	0.5556	0.1278	0.2301

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bache ko samjhao

Socho tum balloon lekar daud rahe ho. Agar tum apne aage ki hawa ko achanak rokdo, woh pile up hoti hai — thodi si geeli, thodi si squishier, aur zyada push karti hai. Jitna tezi daudoge (bada Mach number), utna zyada pile up hoga jab ruka jaye. "Ruke hue" values $T_{0}, P_{0}, ρ_{0}$ hain. Flowing values hamesha chhoti hoti hain, aur kitni chhoti hain ye sirf tumhari speed aur speed of sound ke comparison par depend karta hai.

Active-recall flashcards

Stagnation state define karne wali conditions kya hain?

Flow ko zero velocity tak isentropically decelerate karo (adiabatic + reversible).

T_{0} / T

ka formula

M

ke terms mein?

T_{0} / T = 1 + \frac{γ - 1}{2} M^{2}

P_{0} / P

ka formula?

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

ρ_{0} / ρ

ka formula?

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

T, ρ, P ke exponents (air, γ=1.4)?

1, 2.5, 3.5.

T_{0}

ek adiabatic flow mein friction ke saath bhi constant kyu rehta hai?

Stagnation enthalpy conserved hoti hai (energy eq.); lekin agar irreversible hai to

P_{0}

girta hai.

Air ke liye critical pressure ratio

P^{*} / P_{0}

1. 2^{- 3.5} \approx 0.528

Kaun sa ratio M ke saath sabse tezi se girta hai aur kyu?

Pressure, kyunki iska exponent

γ / (γ - 1) = 3.5

sabse bada hai.

Kya tum ye tables ek normal shock ke across use kar sakte ho?

Nahi — entropy badhti hai,

P_{0}

conserved nahi hota; dono sides ko alag treat karo.

Diya hai

P / P_{0} = 0.3

, air ke liye M?

Solve karo

(1 + 0.2 M^{2})^{3.5} = 1/0.3 \Rightarrow M \approx 1.42

Connections

Speed of Sound and Mach Number — $a = γ R T$ define karta hai jo Step 1 mein use hua.
Stagnation Properties and Energy Equation — $h_{0} =$ const ka origin.
Normal Shock Relations — jahan ye isentropic tables break karte hain.
Converging-Diverging Nozzle & Choking — critical ratios $P^{*} / P_{0}$ , $T^{*} / T_{0}$ use karta hai.
Area-Mach Relation A/A* — usi stagnation framework par bana hai.
Isentropic Process Relations for Perfect Gas — $P \propto T^{γ / (γ - 1)}$ deta hai.