Shock wave angle, deflection angle — Hinglish notes

3.1.14 · Physics › Compressible Flow & Aerodynamics

YEH do angles KYA hain?

Velocity ko decompose KYUN karein? (pehle principles se)

Key trick yeh hai: ek oblique shock sirf ek normal shock hai us velocity component ke liye jo usके perpendicular hai, aur tangential component unchanged rehti hai.

Geometry set up karein:

$u_{1} = V_{1} sin β$ — upstream normal component
$w_{1} = V_{1} cos β$ — tangential component (conserved: $w_{2} = w_{1}$ )
Shock ke peeche flow $θ$ se mur chuki hai, toh normal component downstream velocity se $(β - θ)$ ka angle banata hai: $u_{2} = V_{2} sin (β - θ), w_{2} = V_{2} cos (β - θ)$

Figure — Shock wave angle, deflection angle

$θ$ – $β$ – $M$ relation KAISE derive karein

Step 1 — Normal Mach numbers. Upstream Mach number ka normal component hai $M_{n 1} = M_{1} sin β$ Yeh step kyun? Perpendicular flow ek normal shock ki tarah behave karta hai, aur ek normal shock sirf normal Mach number ki "parwah" karta hai.

Step 2 — Normal-shock density (continuity) ratio use karein. Ek normal shock ke liye density ratio standard Rankine–Hugoniot result hai $\frac{ρ _{2}}{ρ _{1}} = \frac{( γ + 1 ) M _{n 1}^{2}}{( γ - 1 ) M _{n 1}^{2} + 2}$ Yeh step kyun? Shock face ke across mass conservation sirf normal velocity use karta hai: $ρ_{1} u_{1} = ρ_{2} u_{2}$ , isliye $\frac{ρ _{2}}{ρ _{1}} = \frac{u _{1}}{u _{2}}$ .

Step 3 — Velocity ratio ko angles se express karein. Har side par normal ko tangential se divide karein (tangential equal hai): $tan β = \frac{u _{1}}{w _{1}}, tan (β - θ) = \frac{u _{2}}{w _{2}} = \frac{u _{2}}{w _{1}}$ Yeh step kyun? Yeh velocity ratios ko angle ratios mein badal deta hai, jo hum chahte hain. Divide karne par, $\frac{t a n ( β - θ )}{t a n β} = \frac{u _{2}}{u _{1}} = \frac{ρ _{1}}{ρ _{2}} = \frac{( γ - 1 ) M _{n 1}^{2} + 2}{( γ + 1 ) M _{n 1}^{2}}$

Step 4 — $M_{n 1} = M_{1} sin β$ substitute karein aur simplify karein ( $tan = sin / cos$ aur trig identities use karke). Standard compact form hai:

Relation padhna (80/20 essentials)

Common mistakes (Inhe steel-man karein)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum itni tez daud rahe ho ki tumhari warning ki "honk" tumse pehle wall tak nahi pahunch sakti. Jab supersonic air kisi wedge se takraati hai, usse koi warning nahi milti, toh woh ek sharp line ke saath ek saath crash karke mur jaati hai — woh line hai shock. Line ka jhukav hai wave angle. Air ka raasta kitna mura hai woh hai turn angle. Neat trick yeh hai: air ki speed ko "line mein" aur "line ke saath" tood lo. Sirf "line mein" waala hissa squash hota hai; "line ke saath" waala hissa bina change ke nikal jaata hai. Thodi si turning se line leaning rehti hai aur air tez rehti hai; zyada muro toh air surrender kar deti hai — shock pop off ho jaata hai aur ek curved cushion ki tarah aage float karta hai.

Flashcards

Wave angle

β

kya measure karta hai?

Oblique shock aur upstream flow direction ke beech ka angle.

Deflection angle

θ

kya measure karta hai?

Woh angle jitna streamline shock ke across mur jaata hai (= wedge half-angle).

Tangential velocity oblique shock ke across kyun unchanged rehti hai?

Inviscid shock pressure sirf uski face ke perpendicular kaam karta hai, isliye koi tangential force nahi hota ⇒ tangential momentum (aur velocity) conserve hoti hai.

Normal upstream Mach number kya hai?

M_{n 1} = M_{1} sin β

θ

–

β

–

M

relation batao.

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

β

ki lower aur upper limits kya hain?

Lower = Mach angle

μ = sin^{- 1} (1/ M_{1})

; upper =

9 0^{\circ}

(normal shock). Dono

θ = 0

dete hain.

θ < θ_{m a x}

ke liye kitne shock solutions exist karte hain?

Do — ek weak shock (chhota

β

, flow supersonic rehti hai) aur ek strong shock (bada

β

, flow subsonic ho jaati hai).

Agar required

θ > θ_{m a x}

ho toh kya hota hai?

Koi attached oblique shock nahi hota; body ke aage ek detached curved bow shock ban jaata hai.

Oblique shock ke peeche

M_{2}

kaise nikaalte hain?

M_{2} = M_{n 2} / sin (β - θ)

, jahan

M_{n 2}

M_{n 1}

par apply ki gayi normal-shock relation se aata hai.

Relation mein

θ = 0

set karne par kya milta hai?

sin β = 1/ M_{1}

, yaani

β =

Mach angle (ek Mach wave).

Connections

Normal Shock Waves — perpendicular-component limit ( $β = 9 0^{\circ}$ ).
Mach Angle and Mach Waves — $θ \to 0$ lower bound.
Rankine-Hugoniot Relations — density/pressure jumps jo Step 2 mein use hue.
Prandtl-Meyer Expansion — opposite case (flow door mur rahi hai, koi shock nahi).
Detached Bow Shock — $θ_{m a x}$ ke baad kya hota hai.
Supersonic Wedge & Cone Flow — inhi angles ka direct application.