Hypersonic flow — Mach 5+, high temperature effects: Visual walkthrough

3.1.27 · D2 · Physics › Compressible Flow & Aerodynamics › Hypersonic flow — Mach 5+, high temperature effects

Neeche sab kuch scratch se derive kiya gaya hai. Agar koi symbol aata hai, pehle define kiya gaya hai.

Step 1 — Air ka ek parcel, aur "fast" ka matlab kya hai

KYA HAI. Ek chhota sa air ka cube imagine karo — ek parcel — jo stream mein ek wall ki taraf drift kar raha hai. Uski ek speed $V$ hai (har second kitne metres move karta hai) aur ek temperature $T$ hai (uske molecules kitni zor se jagah par jhanjhanate hain).

KYUN. Isse pehle ki hum kahein "fast air gets hot," humein fast ke liye ek seedha measuring stick chahiye. Woh stick hai speed of sound $a$ : woh speed jis par ek chhota sa pressure nudge gas mein travel karta hai. $V$ ko $a$ se compare karna batata hai ki air kisi disturbance se "raste se hat" sakti hai ya nahi waqt par.

PICTURE.

Ek ideal gas ke liye speed of sound hai $a^{2} = γ R T,$ jahan $R$ gas constant hai (ek diye gaye gas ke liye fixed number) aur $γ$ (gamma) hai heat-capacity ratio — abhi ke liye bas "thandi air ke liye 1.4 ke aaspaas ek number." Iska poora matlab Step 6 mein milega.

Step 2 — Parcel ko rokna: motion kahan jaata hai?

KYA HAI. Parcel body se takrata hai aur bilkul nose par ruk jaata hai — stagnation point. Uski speed $V$ se $0$ ho jaati hai.

KYUN. Energy gayab nahi ho sakti. Steady flow ke liye bookkeeping rule, jahan koi heat add nahi aur koi machine kaam nahi kar rahi, yeh hai ki total enthalpy streamline ke saath constant rehta hai: $h + \frac{1}{2} V^{2} = h_{0} .$

$h$ — enthalpy, parcel ki internal energy content per kilogram (uska "thermal wallet").
$\frac{1}{2} V^{2}$ — kinetic energy per kilogram ("motion wallet").
$h_{0}$ — total, stagnation enthalpy, jo fixed hai. Subscript $0$ ka matlab hai "rokne ke baad."

Toh jaise $V$ zero hota hai, motion wallet thermal wallet mein khaali hota hai. Parcel ka $h$ (aur isliye uska temperature) zaroor badhega.

PICTURE.

Step 3 — Enthalpy ko temperature mein badalna

KYA HAI. Hum wallet language ( $h$ ) ko thermometer language ( $T$ ) mein convert karte hain.

KYUN. Ek calorically perfect gas ke liye — jisme heat capacity ek fixed number ho — enthalpy aur temperature simply proportional hote hain: $h = c_{p} T .$

$c_{p}$ — specific heat at constant pressure: temperature ka har degree kitna enthalpy khareedta hai.

Hum yeh isliye use karte hain kyunki humein temperature ke baare mein jawaab chahiye, aur $c_{p}$ wallet ( $h$ ) aur thermometer ( $T$ ) ke beech exchange rate hai. Step 2 ke balance mein substitute karo: $c_{p} T + \frac{1}{2} V^{2} = c_{p} T_{0} .$

$T$ — parcel ka temperature jab woh abhi bhi chal raha tha.
$T_{0}$ — stagnation temperature, poora rokne ke baad uska temperature.

PICTURE.

$\frac{1}{2} V^{2}$ ka har joule $T$ ko $T_{0}$ ki taraf push karta hai exchange rate $c_{p}$ par.

Step 4 — Clean ratio mein rearrange karna

KYA HAI. Ratio $T_{0} / T$ alag karo. Poori equation ko $c_{p} T$ se divide karo: $\frac{T _{0}}{T} = 1 + \frac{V ^{2}}{2 c _{p} T} .$

KYUN. Ratio $T_{0} / T$ dimensionless hai — koi units nahi — toh yeh hume pure number $M$ swap karne dega. Alag " $1$ " sirf original temperature hai jo account ki gayi; doosra term bonus heat hai rокne se.

$\frac{T _{0}}{T}$ — "rokne ke baad kitna guna zyada garam."
$1$ — woh temperature jo tum lekar shuru hue the, saath chali.
$\frac{V ^{2}}{2 c _{p} T}$ — extra heat ka fraction, saari motion energy thermal content se divided.

PICTURE.

Hum Mach number se ek substitution door hain.

Step 5 — Speed of sound fold in karna → master formula

KYA HAI. Step 1 ke do facts use karke $c_{p}$ aur $T$ replace karo.

KYUN. Hum chahte hain $M = V / a$ appear ho, toh hum messy term ko $a^{2} = γ R T$ aur identity $c_{p} = \frac{γ R}{γ - 1}$ se rewrite karte hain (yeh sirf $γ$ ki definition rearranged hai; hum Step 6 mein prove karte hain). Dekho term kaise transform hota hai:

= \frac{V^2(\gamma-1)}{2\gamma R T} = \frac{\gamma-1}{2}\cdot\frac{V^2}{\gamma R T} = \frac{\gamma-1}{2}\cdot\frac{V^2}{a^2} = \frac{\gamma-1}{2}M^2.$$ - Humne $c_p \to \dfrac{\gamma R}{\gamma-1}$ swap kiya, jisse $(\gamma-1)$ upar aa gaya. - Humne $\gamma R T = a^2$ recognize kiya, toh $\dfrac{V^2}{a^2} = M^2$ appear hua. **PICTURE.** > [!formula] Stagnation temperature ratio > $$\boxed{\dfrac{T_0}{T} = 1 + \dfrac{\gamma-1}{2}\,M^2}$$ > - $M^2$ — heating **explode** hone ki wajah: Mach number double karo aur heating term > *chaar guna* ho jaata hai. > - $\dfrac{\gamma-1}{2}$ — "fill rate," jo thandi air ($\gamma=1.4$) ke liye $0.2$ hai. > [!example] $M=10$, $\gamma=1.4$ plug in karo > $$\frac{T_0}{T} = 1 + 0.2\times 100 = 21.$$ > $220\ \text{K}$ stratosphere se shuru karke: $T_0 = 21\times 220 \approx 4620\ \text{K}$. > Yeh *calorically-perfect prediction* hai — isko yaad rakho, Step 7 isko giraa dega. Yahi result [[Rankine–Hugoniot Relations|shock relations]] aur [[Normal and Oblique Shock Waves|shock-wave]] analyses post-shock state ke liye feed karte hain. --- ## Step 6 — $\gamma$ kahan se aata hai: energy ke chhupaane ki jagahon ko count karna **KYA HAI.** Humne $\gamma$ ko "lagbhag 1.4" treat kiya. Ab hum isse earn karte hain. Ek molecule jiggle-energy ko kai independent tareekon se store kar sakta hai jinhein ==degrees of freedom== kehte hain, $f$ se count kiya jaata hai. Equipartition (energy equally share hoti hai) internal energy $e = \tfrac{f}{2}RT$ deta hai, aur isse follow hota hai: $$c_v = \frac{f}{2}R,\qquad \gamma = 1 + \frac{2}{f}.$$ **KYUN.** Yeh poore page ka hinge hai. $\gamma$ **nature ka constant nahi hai** — yeh ek count hai. Zyada storage drawers $f$ ⇒ chhota $\gamma$. Thandi air 3 translation + 2 rotation drawers use karti hai, toh $f=5$ aur $\gamma = 1 + 2/5 = 1.4$. Jab heat *vibration* drawer kholti hai, $f$ 7 ki taraf badhta hai aur $\gamma$ $1.29$ ki taraf girta hai. - $f$ — active energy drawers ki sankhya. - $\tfrac{f}{2}RT$ — har drawer mein $\tfrac12 RT$ energy per mole hoti hai. - $\gamma = 1 + 2/f$ — drawers kholne par girta hai. **PICTURE.** > [!mistake] "γ bas 1.4 hai, toh har jagah 1.4 plug in karo." > Yeh sahi kyun lagta hai: supersonic flow ke liye kaam kiya. Fix yeh hai: hypersonics ke door temperatures ke upar, vibration aur phir chemistry naye drawers kholti hai, $f$ badhta hai, aur $\gamma$ $1.1$–$1.3$ ki taraf slide karta hai. Clean formula ne *fixed* drawers ki sankhya assume ki thi. --- ## Step 7 — Correction: woh heat jo chhup jaaye kabhi thermometer par nahi dikhti **KYA HAI.** $T_0$ ko roughly $800\ \text{K}$ se upar push karo aur molecules **vibrate** karne lagte hain; $\sim 2500\ \text{K}$ ke upar oxygen molecules **dissociate** ho jaate hain (bonds toot jaate hain). Dono energy *soak up* karte hain. [[Real Gas Thermodynamics & Dissociation|Real-gas]] ladder dekho. **KYUN.** Total enthalpy $h_0$ abhi bhi conserved hai — Step 2 kabhi nahi tuta. Lekin ab $h_0$ *thermal* jiggling (jo thermometer read karta hai) aur *hidden* energy (vibration + toote hue bonds) ke beech split ho gaya hai. Kyunki kuch energy hidden pool mein jaati hai, **kam** bacha hai $T$ badhane ke liye. Toh real $T_0$ Step 5 ke calorically-perfect $4620\ \text{K}$ se **kam** hai. $$\underbrace{h_0}_{\text{fixed total}} = \underbrace{c_p T_0}_{\text{thermometer par dikhta hai}} + \underbrace{e_\text{vib} + e_\text{chem}}_{\text{hidden — koi temperature rise nahi}}.$$ **PICTURE.** > [!intuition] Endothermic = ek heat sponge > Ek $\text{O}_2$ bond todna energy *kharach* karta hai, seedha thermal pool se. Toh chemistry ek sponge ki tarah kaam karta hai jo gas ko perfect-gas guess se thanda rakhta hai — yahi precise wajah hai ki [[Boundary Layers & Aerodynamic Heating|heat-shield]] design $\gamma=1.4$ tables use nahi kar sakta. --- ## Step 8 — Edge cases: formula ko uske extremes par check karna **KYA HAI.** Ek achhe formula ka extremes par push karne par sensible behavior hona zaroori hai. **KYUN.** Corners test karna hi middle par trust dilata hai. 1. **$M \to 0$ (mushkil se chal raha):** $\dfrac{T_0}{T} = 1 + 0 = 1$. Ek still parcel ko rokne se koi heat add nahi hoti. ✓ Sensible. 2. **$M \to \infty$ (calorically perfect, fixed $\gamma$):** $T_0/T$ bina bound ke badhta jaata hai — formula infinite temperature kehta hai. Yeh loud warning hai ki fixed $\gamma$ yahan *fiction* hai; Step 7 ke hidden drawers bilkul wahi hain jo reality ko infinity par jaane se rokta hai. 3. **$\gamma \to 1$ (saare drawers khule, $f\to\infty$):** fill-rate $\tfrac{\gamma-1}{2}\to 0$, toh $T_0/T \to 1$. Ek gas jisme infinitely many hiding places hain woh almost garm hi nahi hoti. ✓ Step 7 ke intuition se perfectly match karta hai. **PICTURE.** --- ## Ek-picture summary Poora page ek frame mein: motion energy $\tfrac12V^2$ ek shock se thermal pool mein dalti hai ($T_0$ ko $M^2$ ke saath badhati hai), lekin hypersonic door ke upar uska kuch hissa vibration aur toote hue bonds mein leak ho jaata hai, toh true temperature curve calorically-perfect line ke *neeche* chali jaati hai. > [!recall]- Feynman retelling — seedhe words mein bolo > Socho ek chhota sa air ka cube ek wall ki taraf dauda raha hai. Speed "motion account" mein paisa hai; temperature "heat account" mein paisa hai. Jab cube nose par bilkul ruk jaata hai, rule yeh hai ki uska total paisa nahi badal sakta, toh motion ka har paisa heat account mein aa jaata hai — air garam ho jaati hai. "Kitni tez hai tez" measure karne ke liye, hum uski speed ko speed of sound se compare karte hain; woh ratio hai Mach number $M$. Dhyan se bookkeeping karne par, temperature $1 + \tfrac{\gamma-1}{2}M^2$ se multiply hoti hai, aur kyunki $M$ *squared* hai, tez jaana bilkul jalaa dene wala garam kar deta hai. Lekin $\gamma$ sirf ek count nikalta hai un drawers ka jahan ek molecule energy stash kar sakta hai. Jab air kaafi garam hoti hai, molecules vibrate karne lagte hain aur phir toot jaate hain, naye drawers kholte hain. Jo energy un drawers mein chali jaati hai woh kabhi thermometer par nahi dikhti, toh real air clean formula se thandi rehti hai — aur wahi gap poori wajah hai kyun spacecraft ko heat shields chahiye. > [!recall]- Quick self-test > Heating $M^2$ ke saath kyun scale hoti hai $M$ ke saath nahi? ::: Kyunki stored energy jo heat mein dump honi chahiye woh kinetic energy $\tfrac12 V^2$ hai, jo speed ke *square* par depend karti hai; speed of sound ke through express karne par yeh $M^2$ term ban jaata hai. > Real temperature ko high Mach par $1+\tfrac{\gamma-1}{2}M^2$ follow karne se physically kya rokta hai? ::: Vibrational excitation aur dissociation naye degrees of freedom kholte hain jo energy absorb karte hain (γ girta hai), toh kam energy temperature badhati hai. > Limit $\gamma\to 1$ mein, temperature ratio ka kya hota hai aur kyun? ::: Yeh 1 approach karta hai — infinitely many storage drawers ($f\to\infty$) matlab almost koi energy temperature mein nahi jaati. --- ### Connections Built on: [[Stagnation Properties & Isentropic Relations]], [[Supersonic Flow & Area-Mach Relations]]. Feeds: [[Normal and Oblique Shock Waves]], [[Rankine–Hugoniot Relations]], [[Real Gas Thermodynamics & Dissociation]], [[Boundary Layers & Aerodynamic Heating]]. Parent: [[Hypersonic flow — Mach 5+, high temperature effects]].