Normal shock waves — Rankine-Hugoniot relations (all 5) — derivations: Foundations

3.1.11 · D1 · Physics › Compressible Flow & Aerodynamics › Normal shock waves — Rankine-Hugoniot relations (all 5) — de

Rankine–Hugoniot ki ek bhi line padhne se pehle, tumhe uske har ek symbol ka maalik hona chahiye. Yeh page har ek ko zero se build karta hai, us order mein jis order mein woh ek doosre par depend karte hain, aur peeche ki picture bhi dikhata hai.

1. Flow aur uske do states — subscripts $1$ aur $2$

Gas ek unit area wali tube mein left→right flow karti hai (hum sochtey hain $1 m^{2}$ cross-section taaki hume area kabhi likhna na pade — woh everywhere cancel ho jaata hai).
State 1 = "pehle" ka snapshot; state 2 = "baad" ka snapshot.
Shock khud itni patli hai (kuch molecule-widths) ki hum isse ek flat mathematical surface maante hain jiska koi thickness nahi.

Yeh topic isko kyun chahta hai: har relation ek ratio hai jaise $p_{2} / p_{1}$ — "baad" aur "pehle" ka comparison. Bina dono labels ke koi comparison hi nahi.

2. Density $ρ$ — gas kitni tightly packed hai

Left box (state 1) mein kuch dots hain — low density. Right box (state 2) mein wahi dots zyada paas hain — higher density. Shock gas ko compress karti hai, isliye $ρ_{2} > ρ_{1}$ .
Yeh topic isko kyun chahta hai: "flow compress hoti hai" ka matlab hi " $ρ$ badhta hai" hai. Ratio $ρ_{2} / ρ_{1}$ paanch headline results mein se ek hai.

3. Velocity $u$ — gas kitni tez chal rahi hai

Section 1 ki picture mein, $u_{1}$ lamba arrow hai (tez) aur $u_{2}$ chhota arrow hai (dheema). Shock cross karte waqt, flow slow ho jaati hai: $u_{2} < u_{1}$ .
Yeh topic isko kyun chahta hai: kinetic energy $\frac{1}{2} u^{2}$ aur momentum $ρ u^{2}$ dono $u$ par depend karte hain, aur $u_{1} / u_{2}$ = $ρ_{2} / ρ_{1}$ nikalta hai (Section 8).

4. Pressure $p$ aur temperature $T$

Shock ke paar dono jump karte hain: $p_{2} > p_{1}$ aur $T_{2} > T_{1}$ . Re-entry heating ki kahani $T_{2} ≫ T_{1}$ hai.
Yeh topic isko kyun chahta hai: $p$ momentum push carry karta hai; $T$ thermal energy carry karta hai. Paanch results mein se do $p_{2} / p_{1}$ aur $T_{2} / T_{1}$ hain.

5. Ideal-gas link $p = ρR T$

Yeh topic isko kyun chahta hai: yeh teeno state variables ke beech convert karta hai. Yeh woh bridge hai jo Relations 3 aur 4 se Relation 5 (temperature) nikaalney ke liye use hota hai.

( $p_{0}$ , $T_{0}$ ke gehre meaning ke liye dekho Stagnation properties.)

6. Speed of sound $a$ aur Mach number $M$

$M$ kyun, aur raw $u$ kyun nahi, poore chapter ka star hai? Kyunki ek shock exist bhi kar sakti hai ya nahi yeh sound se flow ki comparison par depend karta hai. Ek pressure signal (ek "warning") $a$ speed se travel karta hai. Agar $u > a$ ( $M_{1} > 1$ ), gas apni khud ki warnings se aage nikal jaati hai — downstream fluid "sun" nahi sakta obstacle ko, toh nature ek wall banaata hai: shock. Isliye paanch relations saari sirf $M_{1}$ ke functions ki tarah likhi jaati hain.

Figure ek source dikhata hai: $M < 1$ par ripples har jagah phailti hain (warning aage pahunch jaati hai); $M > 1$ par woh ek cone/wall mein pile ho jaati hain (upstream koi warning nahi).

( $a$ aur $M$ par zyada: Speed of sound and Mach number.)

7. Heat-capacity ratio $γ$ aur enthalpy $h$

Yeh topic isko kyun chahta hai: energy conservation law padhta hai $h_{1} + \frac{1}{2} u_{1}^{2} = h_{2} + \frac{1}{2} u_{2}^{2}$ . $γ$ aur $h$ -form dono har derivation step mein appear karte hain.

8. Teen conservation laws — poora engine

Saade alfazon mein padho:

Mass: har second jo mass slab mein enter karta hai wahi nikalna chahiye — molecules appear ya vanish nahi hote. Isliye $ρ u$ (mass through unit area per second) dono sides par same hai.
Momentum: flow ka momentum sirf dono faces ke pressure pushes ki wajah se change hota hai. Isliye $p + ρ u^{2}$ (pressure push + momentum flux) across match karta hai.
Energy: koi heat add nahi, bahar koi kaam nahi ⇒ total enthalpy $h + \frac{1}{2} u^{2}$ conserved hai.

Yeh topic isko kyun chahta hai: sab kuch — paanch Rankine–Hugoniot relations saari — in teen lines plus $p = ρR T$ par pure algebra hai. Kuch aur enter nahi karta.

9. Entropy $s$ — woh referee jo real shock choose karta hai

Yeh topic isko kyun chahta hai: conservation laws akele nahi bata sakte ki jump kis direction mein jaayega. Entropy tiebreaker hai, aur yeh yeh bhi explain karta hai kyun total pressure $p_{0}$ girta hai (dekho Stagnation properties).

10. Notation jo milegi — quick glossary

Symbol	Bolte kaise hain	Matlab	Units
$ρ$	"rho"	density	$kg/m^{3}$
$u$	"you"	flow speed	$m/s$
$p$	"pee"	static pressure	$Pa$
$T$	"tee"	static temperature	$K$
$a$	"ay"	speed of sound	$m/s$
$M$	"Mach"	$u / a$	—
$γ$	"gamma"	$c_{p} / c_{v}$	—
$h$	"aitch"	enthalpy	$J/kg$
$R$	"arr"	gas constant	$J/(kg\cdotK)$
$s$	"ess"	entropy	$J/(kg\cdotK)$
$p_{0}, T_{0}$	"pee-nought"	stagnation values	Pa, K
$_{1},_{2}$	—	shock se pehle / baad	—

11. Foundations topic ko kaise feed karte hain

Teen laws (mass, momentum, energy) + ideal-gas link paanch ratios produce karte hain; Mach number woh single variable hai jis par sab depend karte hain; entropy physically real branch select karti hai.

Equipment checklist

Right side cover karo aur khud test karo — tum ready ho jab har line instant ho.

ρ

kya measure karta hai, aur kis units mein?

Mass per unit volume,

kg/m^{3}

u

aur

a

mein kya fark hai?

u

= bulk flow speed;

a = γ R T

= speed jis par ek pressure ripple travel karta hai.

Mach number define karo aur batao

M > 1

ka matlab kya hai.

M = u / a

;

M > 1

matlab supersonic (flow sound se aage nikal jaati hai).

Temperature kelvin mein kyun, Celsius mein kyun nahi?

Gas laws ko ek absolute scale chahiye jo zero se shuru ho.

Ideal-gas equation of state batao.

p = ρR T

Enthalpy

h

kya bundle karta hai, aur uska formula?

Thermal energy + flow push-work;

h = c_{p} T = \frac{γ}{γ - 1} \frac{p}{ρ}

Shock ke paar teen conservation laws likho.

ρ_{1} u_{1} = ρ_{2} u_{2}

;

p_{1} + ρ_{1} u_{1}^{2} = p_{2} + ρ_{2} u_{2}^{2}

;

h_{1} + \frac{1}{2} u_{1}^{2} = h_{2} + \frac{1}{2} u_{2}^{2}

Subscripts 1 aur 2 kya label karte hain?

1 = upstream (shock se pehle, supersonic); 2 = downstream (baad, subsonic).

Do math solutions mein se real shock kaun sa physical law choose karta hai?

Second Law — entropy kam nahi ho sakti.

Paanch relations saari kis ek number par depend karti hain?

Upstream Mach number

M_{1}

Aage: jab har symbol earn ho jaaye, toh paanch ratios banane jaao Normal shock waves — Rankine-Hugoniot relations (all 5) — derivations mein. Related roads: Isentropic flow relations, Oblique shock waves, Rayleigh & Fanno flow.