Detached bow shock — Hinglish notes

3.1.15 · Physics › Compressible Flow & Aerodynamics

YEH HAI KYA?

Key players:

$θ$ — woh deflection jo body flow se maangti hai (jaise wedge/cone ka half-angle, ya blunt nose par effectively $90°$ ).
$θ_{ma x}$ — ek given upstream Mach number $M_{1}$ par attached oblique shock se mila sabse bada turn.

YEH DETACH KYU HOTA HAI? (first principles)

Shuru karo oblique-shock geometry se. Maano $β$ shock-wave angle hai aur $θ$ uske peeche flow ka deflection hai. Upstream velocity ko shock ke normal ( $u_{n}$ ) aur tangential ( $u_{t}$ ) components mein decompose karo.

$θ$ – $β$ – $M$ relation ki derivation (taaki tum dekh sako $θ_{ma x}$ kahan se aata hai):

Upstream normal Mach number: $M_{n 1} = M_{1} sin β$ . Kyun? Shock sirf apne perpendicular component ko "dekhti" hai, aur woh component, sound speed $a_{1}$ se divide karke, $M_{1} sin β$ hota hai.
Ek normal shock ke across, density ratio hai $\frac{ρ _{2}}{ρ _{1}} = \frac{( γ + 1 ) M _{n 1}^{2}}{( γ - 1 ) M _{n 1}^{2} + 2} .$ Kyun? Shock ke across mass + momentum + energy exactly yahi deta hai (Rankine–Hugoniot result).
$u_{t}$ conserved use karke pehle/baad ke velocity triangles ki geometry: $\frac{t a n ( β - θ )}{t a n β} = \frac{u _{n 2}}{u _{n 1}} = \frac{ρ _{1}}{ρ _{2}} .$ Kyun? $tan β = u_{n 1} / u_{t}$ aur $tan (β - θ) = u_{n 2} / u_{t}$ ; divide karo, $u_{t}$ cancel ho jaata hai, aur mass conservation deta hai $u_{n 2} / u_{n 1} = ρ_{1} / ρ_{2}$ .
(2) ko (3) mein substitute karo aur simplify karo standard θ–β–M relation mein: $tan θ = 2 cot β \frac{M _{1}^{2} sin ^{2} β - 1}{M _{1}^{2} ( γ + cos 2 β ) + 2}$

BOW SHOCK KAISA BEHAVE KARTA HAI (centreline → edges)

Worked examples

Common mistakes

Flashcards

Shock detach hokar bow shock kab banta hai?

Jab required deflection

θ

, maximum oblique-shock deflection

θ_{ma x} (M_{1})

se zyada ho jaaye.

Bow shock ki centreline par shock-wave angle

β

kya hota hai?

β = 90°

(wahan yeh normal shock ki tarah behave karta hai).

Bow shock ki centreline ke theek peeche flow subsonic hai ya supersonic?

Subsonic (

M_{2} < 1

Stand-off distance kya hota hai?

Detached shock aur body ke nose ke beech ka gap

Δ

M_{1}

badhne par stand-off distance kaise change hota hai?

Yeh chhota hota hai (shock body ko hug karta hai), high Mach par ek chhoti finite value par tend karta hai.

Mathematically shock detach kyun hota hai?

θ–β–M relation ka

θ > θ_{ma x}

ke liye koi real

β

solution nahi hota.

θ–β–M relation likho.

tan θ = 2 cot β \frac{M _{1}^{2} sin ^{2} β - 1}{M _{1}^{2} ( γ + cos 2 β ) + 2}

M_{n 1}

M_{1}

aur

β

ke terms mein kya hota hai?

M_{n 1} = M_{1} sin β

M_{1} = 2

γ = 1.4

par

θ_{ma x}

approximately kya hota hai?

Lagbhag

22.97°

Oblique shock ke across tangential velocity kyun conserved rehti hai?

Shock sirf apne normal ke along act karti hai, isliye tangential component unchanged rehta hai.

Recall Feynman: 12-saal ke bachche ko samjhao

Socho tum bahut tez daud rahe ho aur apne aage ki air ko push kar rahe ho. Agar tumhara shape pointy arrow jaisa hai aur tum super fast ja rahe ho, to air seedha point ke saath fold ho sakti hai — yeh attached shock hai. Lekin agar tumhara front mota aur gol hai, to pile ho rahi air itni sharply bend nahi kar sakti ki tumhari shape follow kar sake. To woh haar maanti hai aur squished air ki ek curved wall bana deti hai jo tumhare theek aage float karti hai, jaise ek invisible bubble shield. Us shield ke bilkul beech mein air l거의 ruk jaati hai aur "sound speed" se neeche aa jaati hai, jisse woh calmly tumhare mote nose ke around slide ho sakti hai. Tez jao aur shield tumhare chehre se aur tight ho jaayegi.

Connections

Oblique shock waves — bow shock inhi ki ek continuous family hai.
Normal shock relations — centreline par exact math.
Maximum deflection angle and weak/strong shock solutions — $θ_{ma x}$ define karta hai.
Mach angle and Mach waves — body se door bow shock ki limit.
Hypersonic flow and shock layers — jahan stand-off distance bahut chhoti ho jaati hai.
Stagnation properties across shocks — nose par peak pressure/temperature.