Hypersonic flow — Mach 5+, high temperature effects

3.1.27 · Physics › Compressible Flow & Aerodynamics

WHAT hai "hypersonic"?

WHY 5 jaisa number? Kyunki shock ke peeche temperature rise $M^{2}$ ke saath scale karta hai. Around $M \approx 5$ , post-shock temperatures ~1000–1500 K se upar chadh jaate hain, jahan oxygen vibration aur phir dissociation matter karne lagti hai. Yeh label physics track karta hai, koi clean threshold nahi.

HOW temperature blow up hoti hai: stagnation temperature derive karna

Derivation from first principles (adiabatic, steady, no work):

Ek streamline ke along energy conservation (steady adiabatic flow) kehta hai ki total enthalpy constant hai: $h + \frac{1}{2} V^{2} = h_{0} = const$

WHY yeh step? Koi heat add nahi, koi shaft work nahi, toh steady-flow energy equation reduce ho jaati hai: "enthalpy + kinetic energy = constant." $h_{0}$ stagnation enthalpy hai.

Ek calorically perfect gas ke liye $h = c_{p} T$ , toh: $c_{p} T + \frac{1}{2} V^{2} = c_{p} T_{0}$

$c_{p} T$ se divide karo: $\frac{T _{0}}{T} = 1 + \frac{V ^{2}}{2 c _{p} T}$

Ab $c_{p} = \frac{γ R}{γ - 1}$ use karo aur speed of sound $a^{2} = γ R T$ , toh $\frac{V ^{2}}{2 c _{p} T} = \frac{V ^{2} ( γ - 1 )}{2 γ R T} = \frac{( γ - 1 )}{2} \frac{V ^{2}}{a ^{2}} = \frac{γ - 1}{2} M^{2}$ .

WHY $γ$ change hota hai: degrees of freedom aur real-gas effects

Equipartition se, per mole internal energy active degrees of freedom $f$ ke saath badhti hai: $e = \frac{f}{2} R T, c_{v} = \frac{f}{2} R, γ = \frac{c _{p}}{c _{v}} = 1 + \frac{2}{f} .$

WHY: zyada active modes $\Rightarrow$ bada $f \Rightarrow$ chhota $γ$ .

Regime	Active modes	Approx $f$	$γ$
Cold air (trans + rot)	3 + 2	5	1.40
Vibration excited (~800 K+)	+ vibration	→7	→1.29
Dissociation (~2500 K+)	bonds break, new species	—	~1.1–1.2
Ionization (~9000 K+)	electrons free	—	varies

HOW shock layer thin hoti hai: Mach number aur density ratio

Ek normal shock ke liye, limiting density ratio jaise $M \to \infty$ (calorically perfect): $\frac{ρ _{2}}{ρ _{1}} \to \frac{γ + 1}{γ - 1}$

WHY: Rankine–Hugoniot relations se, velocity ratio $u_{2} / u_{1} \to (γ - 1) / (γ + 1)$ , aur $ρ u$ conserved hai, isliye density reciprocal se badh jaati hai. $γ = 1.4$ ke liye yeh 6 par cap hoti hai. Real-gas effects se $γ$ gir jaata hai, isliye $ρ_{2} / ρ_{1}$ 15–20 tak pahunch sakti hai — shock aur bhi close.

Figure — Hypersonic flow — Mach 5+, high temperature effects

Newtonian impact theory (hypersonic ka 80/20 shortcut)

Derivation: Density $ρ_{\infty}$ , speed $V_{\infty}$ ki ek stream ek surface se takraati hai jo flow se angle $θ$ par inclined hai. Unit area se takraane wala mass flux $= ρ_{\infty} V_{\infty} sin θ$ . Har parcel normal velocity $V_{\infty} sin θ$ kho deta hai. Pressure = destroyed momentum flux: $p - p_{\infty} = (ρ_{\infty} V_{\infty} sin θ) (V_{\infty} sin θ) = ρ_{\infty} V_{\infty}^{2} sin^{2} θ .$

Define karo $C_{p} = \frac{p - p _{\infty}}{\frac{1}{2} ρ _{\infty} V _{\infty}^{2}}$ :

WHY re-entry ke liye blunt better hai

Hypersonic regime ki shuruaat roughly kaun sa Mach number mark karta hai?

About M ≈ 5 (ek physics-based rule of thumb, koi sharp wall nahi).

"Hypersonic" sirf Mach number ki jagah physics se kyun define hota hai?

Kyunki nayi effects (thin shock layers, viscous interaction, entropy layer, real-gas/high-T chemistry) dominant ho jaati hain — yeh regime uss physics ke baare mein hai jo ab ignore nahi ki ja sakti.

Stagnation temperature ratio derive/state karo.

T_{0} / T = 1 + \frac{γ - 1}{2} M^{2}

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

se.

High Mach par calorically-perfect

T_{0}

formula temperature kyun overpredict karta hai?

Energy vibration, dissociation, aur ionization mein absorb ho jaati hai, isliye

T

kam rehti hai; saath hi

γ

1.4 se neeche gir jaata hai.

Normal-shock density ratio ka calorically-perfect limit kya hai?

(γ + 1) / (γ - 1)

= 6 for

γ = 1.4

; real-gas effects ke saath 15–20 tak badh jaata hai.

Hypersonic flow mein shock layer thin kyun ho jaati hai?

Shock ke across high density ratio same mass ko ek thin layer mein compress kar deta hai, real-gas

γ

reduction se aur thin ho jaati hai.

Newtonian impact theory ka pressure coefficient state karo aur uski origin derive karo.

C_{p} = 2 sin^{2} θ

; impacting particles ke destroyed normal momentum flux

ρ_{\infty} V_{\infty}^{2} sin^{2} θ

se.

Mach-independence principle kya hai?

High Mach par, force coefficients (Newtonian theory ke zariye) body geometry/inclination par depend karte hain, Mach number par zyada nahi.

Air ke liye high-temperature ladder list karo jaise T badhti hai.

Vibration (~800 K) → O₂ dissociation (~2500 K) → N₂ dissociation (~4000 K) → ionization (~9000 K, plasma/blackout).

γ

ko degrees of freedom

f

se relate karo.

γ = 1 + 2/ f

; zyada active modes → bada

f

→ chhota

γ

Re-entry vehicles blunt kyun hote hain, sharp kyun nahi?

Bluntness bow shock ko detach kar deta hai aur heat air mein dump karta hai; stagnation heat flux

\propto 1/ R_{n}

, toh bada nose radius heating kam karta hai.

Re-entry communications blackout ka kya cause hai?

Shock-heated air ka ionization ek plasma form karta hai jo radio waves absorb/reflect karta hai.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho tum itna fast daud rahe ho ki hawa tumhare raaste se hat hi nahi sakti — woh tumhare aage ek patli invisible wall mein pile up ho jaati hai (ek shock). Super fast jaane par (sound ki speed se 5 guna), hawa ko squash karna use incredibly hot bana deta hai, jaise haath ragadna lekin hazaar guna zyada. Itni zyada hot ho jaati hai ki hawa ke molecules kaampe lagte hain, phir snap ho jaate hain, phir Sun ke ek tiny piece ki tarah glow karne lagte hain. Yahi heat hai isliye Earth par wapas aane wale spaceships ko shields chahiye hote hain, aur isliye hum front ko round aur fat banate hain — ek fat nose burning-hot air ko ship se door push karta hai instead of use skin se touch karne dene ke.

Connections

Normal and Oblique Shock Waves — post-shock $T$ , $ρ$ jumps ka source jo yahan use hue.
Stagnation Properties & Isentropic Relations — $T_{0} / T = 1 + \frac{γ - 1}{2} M^{2}$ ki origin.
Rankine–Hugoniot Relations — $(γ + 1) / (γ - 1)$ density limit deta hai.
Real Gas Thermodynamics & Dissociation — kyun $γ$ girta hai aur energy hide ho jaati hai.
Boundary Layers & Aerodynamic Heating — stagnation heat flux aur TPS design.
Supersonic Flow & Area-Mach Relations — hypersonic se thoda neeche ka regime.