Wave drag — transonic and supersonic — Hinglish notes

3.1.25 · Physics › Compressible Flow & Aerodynamics

1. Wave drag kya hai? (WHAT)

2. $M = 1$ cross karne mein energy kyun lagti hai? (WHY — first principles se derivation)

HOW hum idea build karte hain:

Ek sound wave ek tiny pressure pulse hai jo $a$ par move karta hai. $u$ par move karne wali body pressure pulses bahar bhejti hai. Mach number hai

$M = \frac{u}{a} .$

Imagine karo pulses har ek instant par emit ho rahi hain. Time $t$ mein:

Ek pulse radius $a t$ tak spread ho chuki hai.
Body $u t$ move kar chuki hai.

Agar $u < a$ ( $M < 1$ ): body apne khud ke pulse spheres ke andar rehti hai → aage ki air warn ho jaati hai → smooth subsonic deflection.

Agar $u > a$ ( $M > 1$ ): body apne pulses ko outrun kar leti hai. In infinitesimal pulses ka envelope Mach cone hai, jiska half-angle $μ$ satisfy karta hai

$sin μ = \frac{a t}{u t} = \frac{a}{u} = \frac{1}{M} .$

Formula Mach angle

$sin μ = \frac{1}{M} μ = arcsin (\frac{1}{M})$ Yeh step kyun? Yeh "pulse front radius $a t$ " vs "distance travelled $u t$ " ki pure geometry hai. Jaise $M \to 1$ , $μ \to 90°$ . Jaise $M \to \infty$ , $μ \to 0$ (cone body ke tight wrap ho jaata hai).

CAUTION (important): Mach angle $μ$ $M > 1$ flow mein infinitesimally weak disturbances ("Mach waves") ke envelope ko describe karta hai. Yeh body par actual shock ka angle NAHI hai. Ek real shock ek finite-strength disturbance hai aur hamesha Mach cone se steeper hoti hai:

Ek slender wedge/cone ke liye half-angle $θ$ ke saath, attached oblique shock angle $β$ full == $θ$ – $β$ – $M$ relation== se aata hai aur $β > μ$ satisfy karta hai.
Ek blunt body ke liye (ya $θ$ maximum deflection se upar), shock detach ho jaati hai aur ek curved bow shock ke roop mein khadi rehti hai; centreline ke paas yeh locally ek normal shock hai, kahin bhi $μ$ ke barabar nahi. Toh: Mach cone = weak-signal geometry; bow/oblique shock = finite compression jo shock relations se govern hoti hai, $sin μ = 1/ M$ se nahi.

$θ$ – $β$ – $M$ relation (kyun ek real shock angle $μ$ nahi hai)

Ek oblique shock ke liye jo flow ko angle $θ$ se upstream Mach $M_{1}$ par mod deti hai, shock ke across mass/momentum/energy deta hai

$tan θ = 2 cot β \frac{M _{1}^{2} s i n ^{2} β - 1}{M _{1}^{2} ( γ + c o s 2 β ) + 2} .$

Kyun ek shock = drag (entropy argument)

Ek oblique/normal shock ke across flow adiabatic but irreversible hoti hai, isliye total enthalpy $h_{0}$ conserved rehti hai lekin entropy badhti hai: $s_{2} > s_{1}$ . Ek adiabatic compressible flow ke liye Gibbs se,

$\frac{p _{0, 2}}{p _{0, 1}} = e^{- Δ s / R} .$

Toh har shock ke across stagnation pressure drop hota hai. Ek control volume par momentum balance dikhata hai ki wake mein stagnation-pressure deficit body par ek net rearward force ke barabar hai. Woh force wave drag hai. Koi viscosity nahi chahiye — yeh ek purely compressible, shock-born loss hai.

3. Transonic regime — drag $M = 1$ ke paas spike kyun karta hai

Drag coefficient bump transonic mein bahut bada hota hai, phir pure supersonic mein phir se girta hai kyunki shocks attached, oblique aur per unit length weaker ho jaati hain.

Figure — Wave drag — transonic and supersonic

4. Supersonic regime — quantitative wave drag

Ek thin 2-D airfoil ke liye small angle of attack par supersonic flow mein, linearized (Ackeret) theory surface pressure coefficient ek single oblique-wave relation se deta hai:

$C_{p} = \frac{2 θ}{M _{\infty}^{2} - 1},$

jahan $θ$ flow ke relative local surface inclination hai. (Note: yeh linearized result weak-disturbance limit hai, jahan local wave angles $μ$ ke approach karte hain; yeh full $θ$ – $β$ – $M$ shock/expansion physics ka small- $θ$ approximation hai.)

5. Area Rule (80/20 design win)

6. Common mistakes

7. Active recall

Recall Khud test karo (answers hide karo)

Wave drag bina friction ke kyun exist karta hai? → shock entropy rise ⇒ stagnation-pressure loss ⇒ net rearward force.
Mach angle ka formula? → $sin μ = 1/ M$ (sirf weak-disturbance envelope).
Kya ek real shock angle $μ$ par hota hai? → Nahi; attached shock $β > μ$ via $θ$ – $β$ – $M$ ; blunt body ⇒ detached bow shock.
Supersonic wave drag $M$ ke saath decrease kyun karta hai? → $\propto 1/ M^{2} - 1$ .
Thin swept wings kyun? → $M_{cr}$ raise karo aur thickness wave drag cut karo; $M_{n} = M cos Λ$ .

Recall Feynman: ek 12-saal ke bachche ko explain karo

Imagine karo swimming pool mein dauraana. Slowly, paani tumhare pahunchne se pehle side ho jaata hai. Lekin agar tum ripples se faster dauro, toh paani tumhare raaste se hat nahi sakta — woh tumhare aage ek sharp wall mein slam kar jaata hai. Us paani ki wall ko push karna mushkil kaam hai, aur woh energy kabhi wapas nahi milti. Sound se faster jet ek "air ki wall" banata hai jise shock wave kehte hain, aur use aage push karna wave drag hai. Ek faint, gentle cone of tiny ripples bhi hota hai (Mach cone) — woh woh strong wall nahi hai; real wall (shock) zyada steeply lean karti hai aur push karna bahut mushkil hai. Plane ko thin aur pointy banao toh wall weak ho jaati hai, push karna aasaan ho jaata hai.

Connections

Speed of sound and Mach number
Normal and oblique shock waves
Theta-beta-M relation and detached bow shocks
Entropy and stagnation pressure loss
Critical and drag-divergence Mach number
Prandtl-Glauert compressibility correction
Whitcomb Area Rule and supercritical airfoils
Sonic boom and the Mach cone

Wave drag physically kya hai?

Shock-wave formation se drag; shocks ke across entropy badhti hai ⇒ stagnation-pressure loss ⇒ net rearward force (origin mein inviscid).

Mach angle formula?

sin μ = 1/ M

, toh

μ = arcsin (1/ M)

— sirf infinitesimal disturbances ka envelope.

M=2 par Mach angle?

3 0^{\circ}

Kya body ka shock Mach angle par hota hai?

Nahi. Attached oblique shock angle

β > μ

via

θ

–

β

–

M

relation; blunt bodies (ya

θ > θ_{ma x}

) detached, curved bow shock dete hain (nose par locally normal).

Shock angle

β

Mach angle

μ

ke barabar kab hota hai?

Sirf vanishing turning ki limit mein,

θ \to 0

(zero-strength weak shock).

Shock detach hokar bow shock kyun ban jaata hai?

Required flow turning

θ_{ma x} (M)

se exceed karta hai (e.g. blunt nose) — koi attached oblique-shock solution exist nahi karta.

Critical Mach number

M_{cr}

Free-stream

M

jis par local flow pehli baar

M = 1

tak pahunchti hai.

Drag-divergence Mach

M_{dd}

M_{cr}

M_{dd} > M_{cr}

; yeh woh hai jahan

C_{D}

tezi se badhna shuru karta hai.

Supersonic wave drag Mach ke saath kaise scale karta hai?

c_{d, w a v e} \propto 1/ M^{2} - 1

—

M

badhne par decrease karta hai.

Thin supersonic airfoil ke liye Ackeret

C_{p}

C_{p} = 2 θ / M^{2} - 1

(linearized weak-disturbance limit).

Ackeret lift coefficient (flat plate)?

c_{l} = 4 α / M^{2} - 1

Ackeret wave-drag coefficient?

c_{d} = \frac{4}{M ^{2} - 1} (α^{2} + \overline{(d y_{c} / d x)^{2}} + \overline{(d y_{t} / d x)^{2}})

Wave drag se flat-plate supersonic L/D?

c_{l} / c_{d} = 1/ α = cot α

Wing sweep ka critical Mach par effect?

Use raise karta hai; effective normal Mach

M_{n} = M cos Λ

chhota hota hai, shocks delay hoti hain.

High speed par thin wings kyun?

Thickness term zero-lift wave drag dominate karta hai aur

M_{cr}

lower karta hai.

Area Rule kya hai?

Transonic wave drag minimize karne ke liye axial cross-section area

A (x)

smooth karo (waisted "coke-bottle" fuselage).

Kya wave drag viscous hai?

Nahi — origin mein inviscid; even frictionless gas mein shocks ke through arise karta hai.