Aerodynamic heating during reentry — stagnation point heat flux Chapman equation

3.4.21 · Physics › Rocket Flight Mechanics

1. Stagnation point actually hai kya?

KYA chahiye hamein: heat flux $\overset{q}{˙}$ (units $W/m^{2}$ ) us point par surface mein.

KYU naak par aur sides par kyun nahi? Ek blunt nose ek strong detached bow shock banata hai. Uske peeche gas compress aur superheat ho jaati hai, aur stagnation streamline par poori tarah decelerate hoti hai, isliye wahin dump hone wali enthalpy maximum hoti hai. (Blunt bodies jaanbujhkar use kiye jaate hain: ek sharp nose isko aur bhi zyaada concentrate karta aur pighal jaata.)

2. First principles se scaling derive karna

Hum sirf Chapman ke numbers memorize nahi karte — hum equation ki form banate hain.

Step 1 — Available energy. Gas ki total (stagnation) enthalpy vehicle ke relative $h_{0} = h_{\infty} + \frac{1}{2} V^{2} \approx \frac{1}{2} V^{2}$

Yeh step kyun? Hypersonic speed par ordered kinetic energy $\frac{1}{2} V^{2}$ , ambient thermal enthalpy $h_{\infty}$ ko bahut peeche chhod deti hai, isliye heat "budget" $V^{2}$ ke saath scale karta hai.

Step 2 — Heat boundary layer cross kaise karta hai. Heat wall tak ek patli boundary layer ki thickness $δ$ ke through conduction/diffusion se pahunchti hai. Fourier-style, $\overset{q}{˙} \sim k \frac{T _{0} - T _{w}}{δ} \sim ρ_{e} u_{e} c_{p} \frac{( T _{0} - T _{w} )}{(some Reynolds factor)}$

Yeh step kyun? Driving potential hot stagnation gas aur thandi wall ke beech ki enthalpy difference hai; resistance boundary-layer diffusion length hai.

Step 3 — Boundary-layer thickness scaling. Stagnation region ke liye velocity gradient $β = (d u_{e} / d x)_{s}$ flow set karta hai. Self-similar stagnation boundary layer solve karne par $\overset{q}{˙} \propto ρ_{e} β (h_{0} - h_{w}), β \propto \frac{1}{R _{n}} \frac{p _{0} - p _{\infty}}{ρ} \propto \frac{V}{R _{n}}$

Yeh step kyun? Bada nose radius $R_{n}$ flow ko ek lambi distance par stretch karta hai $\Rightarrow$ gentle gradient $\Rightarrow$ thicker boundary layer $\Rightarrow$ kam heat flux. Isliye blunt noses survive karte hain.

Step 4 — Scaling collect karo. $β \propto V / R_{n}$ , $ρ_{e} \propto ρ_{\infty}$ , aur $(h_{0} - h_{w}) \propto V^{2}$ rakhne par: $\overset{q}{˙} \propto \frac{ρ _{\infty}}{R _{n}} V^{3}$

Yeh $V^{3}$ (teen powers!) aur $1/ R_{n}$ stagnation heating ki pehchaan hain.

3. Heating sabse buri kab hoti hai? (Forecast-then-Verify)

Figure — Aerodynamic heating during reentry — stagnation point heat flux Chapman equation

4. Common mistakes (Steel-manned)

5. Radiative heating footnote

Chapman/Sutton–Graves convective heating cover karta hai. Bahut zyaada speed par ( $V ≳ 10$ km/s, e.g. Mars/lunar return) shocked gas glow karne lagti hai aur radiative flux add hoti hai jo roughly $\overset{q}{˙}_{r a d} \propto R_{n} ρ^{a} V^{b}$ ke saath scale karti hai, $b \sim 8$ –15 ke saath — note karo yeh $R_{n}$ ke saath badhti hai, convective se ulta. Toh super-blunt nose convective relief ke liye radiative penalty leta hai. Yeh tension real heat-shield design ko shape karta hai.

Flashcards

Reentry vehicle par stagnation point kya refer karta hai?

Naak ka woh point jahan incoming flow poori tarah rest mein aa jaaye; maximum temperature, pressure, aur convective heat flux ka jagah.

Chapman/Sutton–Graves stagnation heat-flux formula batao.

\overset{q}{˙}_{s} = \frac{C}{R _{n}} ρ_{\infty} V^{3}

C \approx 1.83 \times 1 0^{- 4}

ke saath (SI, Earth air).

Stagnation heat flux velocity ke saath kaise scale karta hai?

V^{3}

ke roop mein — speed double karne par 8× heating hoti hai.

Stagnation heat flux nose radius ke saath kaise scale karta hai?

R_{n}^{- 1/2}

ke roop mein — bada (blunter) nose flux kam karta hai; 4× radius flux aadha kar deta hai.

Reentry capsules blunt kyun hoti hain, sharp kyun nahi?

Bada

R_{n}

\overset{q}{˙} \propto R_{n}^{- 1/2}

kam karta hai; detached bow shock energy air mein dump karta hai aur boundary layer thick karta hai, surface protect karta hai.

V

ka exponent teen kyun hai, do kyun nahi?

Energy content

V^{2}

ke saath scale karta hai, lekin boundary-layer mass flux ek aur factor

\propto V

add karta hai, transfer rate ke liye

V^{3}

deta hai.

Kya peak heating peak deceleration ke saath coincide karti hai?

Nahi — peak heating zyaada altitude par hoti hai (kam

ρ

); peak g-load gehri jagah hoti hai.

Ballistic peak heating kis density par hoti hai?

ρ_{peak} = β sin γ / (3 H)

, matlab atmosphere mein relatively upar.

Chapman kaunsa heating mode cover karta hai, aur bahut zyaada speed par kya miss hai?

Convective heating; radiative heating (glowing shock gas se) ~10 km/s se upar important ho jaati hai aur

R_{n}

ke saath opposite tarike se scale karti hai.

Recall Feynman: 12-saal ke bachche ko samjhao

Socho apne haath bahut tez ragad rahe ho — woh garam ho jaate hain. Ek wapas aane wala spaceship hawa ke saath hazaaron guna tez ragadhta hai, toh naak ke aage ki hawa aag jaisi garam ho jaati hai. Ekdum tip par, jahan hawa bilkul ruk jaati hai, wahan sabse zyaada garmi hoti hai. Ek gol, moti naak actually ek pointed se thandi hoti hai, kyunki yeh jalti hui garam hawa ko ship se door dhakelta hai jaise ek snowplow. Rule ke liye tip kitni garam hoti hai: yeh speed par bahut strongly depend karta hai (teen baar — thoda tez = BAHUT zyaada garam), thodi si hawa ki thickness par, aur agar naak badi aur gol ho toh thandi ho jaati hai.

Connections

Reentry trajectory dynamics — $V (t)$ aur $ρ (h)$ provide karta hai jo Chapman mein plug karne hote hain.
Ballistic coefficient and deceleration — set karta hai ki peak-heating vs peak-g kahan hoti hai.
Bow shock and blunt-body theory — $R_{n}^{- 1/2}$ dependence ke peeche ki physics.
Boundary layer theory — $ρ u$ heat-transfer factor ka origin.
Thermal Protection Systems (ablatives, tiles) — $\overset{q}{˙}_{s}$ heat shield ki sizing kaise karta hai.
Radiative heating at hypersonic speeds — complementary mode jo Chapman miss karta hai.