Foundations — Wave drag — transonic and supersonic

3.1.25 · D1 · Physics › Compressible Flow & Aerodynamics › Wave drag — transonic and supersonic

Yeh page parent note (Wave drag — transonic and supersonic) mein use hone wale har letter, ratio, aur picture ko fully build karta hai. Upar se neeche padho: koi bhi cheez use hone se pehle explain ki jaayegi.

0. Khel ke characters

Symbols se pehle, physical actors se plain words mein milte hain:

• Hawa — ek springy fluid. Dabao toh push back karta hai; woh push pressure hai.
• Ek disturbance — hawa ko koi bhi nudge (paas se guzarta plane ka nose). Yeh bahar ki taraf pressure ripple ki tarah spread hota hai.
• Body — woh cheez jo hawa mein move kar rahi hai (wing, wedge, nose cone).

Baaki sab kuch in teeno mein se kisi ek se attached ek number hai.

1. Pressure $p$ — "push per area"

Picture: socho ek swarm of tiny air molecules ek wall ko hammer kar rahi hain. Har hit chhoti hai; billions per second ek steady push mein add ho jaate hain. Woh steady push per square metre hi $p$ hai.

Topic ko iski zarurat kyun hai: wave drag ultimately pushes mein difference hai. Agar hawa body ke front pe zyada aur back pe kam push kare, toh bacha hua hissa ek backward force ban jaata hai. Shocks exactly woh regions hain jahan pressure jump karta hai, isliye $p$ drag ka raw material hai.

2. Speed of sound $a$ — ek "warning" kitni tez travel karti hai

Picture (Figure s01): pond mein ek pathar daalo aur rings ek fixed speed pe baahir ki taraf spread hoti hain. Hawa mein bhi yehi 3-D mein hota hai: koi bhi nudge pressure ki ek expanding sphere baahir bhejta hai. Us sphere ki radius time $t$ ke baad $at$ hoti hai.

Exactly yeh tool kyun? Hawa sirf pressure signals idhar udhar paas karke hi "khud ko raaste se hata sakti hai." Woh signals $a$ ki speed se move karte hain — isse tez nahi. Isliye $a$ advance warning ki speed limit hai. Wave drag ka poora drama is baare mein hai ki body is speed limit ko beat karti hai ya nahi. Speed of sound and Mach number dekho.

3. Flow speed $u$ aur Mach number $M$ — crucial comparison

Ab poore chapter ka sabse important number. Hum akele $u$ ya akele $a$ ki parwah nahi karte — hum care karte hain ki kaun bada hai. Do speeds compare karne ka natural tarika hai unhe divide karna:

• $M<1$ → subsonic
• $M=1$ → sonic
• $M>1$ → supersonic
• $M$ roughly 1 ke aas paas (roughly 0.8–1.2) → transonic

Subscript $M_{\infty }$ (padho "M-infinity") ka matlab hai free stream ka Mach number — body se door undisturbed hawa, body ke kuch bigaadne se pehle.

4. Mach cone banana — apni warnings se aage nikal jaane ki geometry

Yeh ab tak ki sab cheez ka payoff hai. Dekho Section 2 ke spheres kya karte hain jab body move karti hai.

Picture (Figure s02): body har instant pe ek pressure sphere emit karti hai.

• Time $t$ mein, ek purana sphere $at$ radius tak grow kar chuka hai.
• Usi time mein, body $ut$ distance aage move kar chuki hai.

Agar $u<a$ ($M<1$): body apne saare spheres ke andar rehti hai. Aage ki hawa ko hamesha warning milti hai. Smooth flow.

Agar $u>a$ ($M>1$): body apne spheres se aage daud jaati hai. Saare spheres body ke peeche ek cone ke saath line up ho jaate hain — Mach cone. Iska front piled-up ripples ki ek diwar hai.

Cone ka angle dhundhna — hume sine kyun chahiye. Figure s02 mein right triangle dekho: body ka path (length $ut$) lambi side (hypotenuse) hai, aur sphere ki radius (length $at$) cone ke half-angle $\mu$ ke opposite side hai. Woh trig ratio jo "angle ke opposite side" ko "hypotenuse" se connect karta hai woh sine hai:

$\sin (\text{angle})=\frac{\text{opposite}}{\text{hypotenuse}}$

Hum sine choose karte hain (tangent ya cosine nahi) precisely isliye kyunki humein opposite side aur hypotenuse pata hain — yeh woh do lengths hain jo geometry humein deti hai. Isliye:

Saare cases:

• $M\to 1^{+}$: $1/M\to 1$, isliye $\mu \to 90°$ — cone flat hokar flow ke seedha across ek diwar ban jaata hai.
• $M=2$: $\sin \mu =1/2$, $\mu =30°$.
• $M\to \infty$: $1/M\to 0$, isliye $\mu \to 0°$ — cone body ke khilaf tight ho jaata hai.
• $M\le 1$: $1/M\ge 1$, aur koi real angle aisa nahi jiska sine 1 se bada ho — equation ka koi solution nahi. Yeh maths bol raha hai ki speed of sound se neeche koi cone nahi hota.

Iske baare mein zyada jaankari Sonic boom and the Mach cone mein.

5. Angle $\theta$, shock angle $\beta$, aur real shock $\mu$ se steep kyun hota hai

Picture (Figure s03): half-angle $\theta$ wala ek wedge. Dashed line $\mu$ pe (weak-signal cone) solid line $\beta$ pe (real attached shock) ke andar baithti hai. Required turn jitna strong, shock utna zyada forward lean karta hai.

$\theta$, $\beta$, aur $M$ ke beech exact link θ–β–M relation hai. Iske extreme cases pe do words taaki baad mein koi surprise na ho:

• Chhota $\theta$ → weak shock, $\beta$ $\mu$ ke paas.
• $\theta$ maximum ${\theta }_{max}$ se bada → koi attached shock exist nahi karta; woh body se ek curved bow shock ke roop mein alag ho jaata hai jo blunt nose ke aage khada hota hai. Dekho Normal and oblique shock waves.

$\gamma$ (gamma), jo us relation mein appear karta hai, ratio of specific heats hai — gas ki ek fixed property (hawa ke liye roughly $1.4$) jo describe karta hai ki squeeze hone par woh kitna strongly heat karta hai. Bas itna jaanna kaafi hai ki yeh $1.4$ ke paas ek constant hai.

6. Entropy $s$, gas constant $R$, aur stagnation pressure $p_0$ — ek shock kyun drag ke barabar hai

Yeh matter kyun karte hain, aur unke beech link. Shock ke across flow apni total energy rakhti hai (kuch bahar heat ke roop mein nahi nikalta) lekin kuch ko disorder mein waste kar deta hai. Exact accounting hai:

Exponential $e^x$ yahan kyun? Yeh woh function hai jo ek additive change (entropy, jo add up hota hai) ko ek multiplicative factor (pressure ratio) mein convert karta hai. Kyunki $\Delta s>0$ hai, exponent negative hai, isliye $e^{-\Delta s/R}<1$: downstream stagnation pressure hamesha upstream ka ek fraction hota hai. Stagnation pressure kho jaata hai.

Aur khoye hue $p_0$ = drag kyun: body ki wake mein uski nigali hawa se kam pushing potential carry hoti hai. Woh missing momentum, Newton ke bookkeeping se body ke around ek box pe, ek net backward force ke roop mein dikhta hai — wave drag, bina kisi friction ke. Poora treatment: Entropy and stagnation pressure loss.

7. Dynamic pressure $q$, pressure coefficient $C_p$, drag coefficient $C_D$, aur factor $\sqrt{M^2-1}$

Drag ko dimensionless banane se pehle humein flow ki "rush" ko ek pressure ki tarah chahiye. Moving air kinetic energy carry karta hai; jab woh surface pe pile karta hai, woh energy extra push ki tarah show hoti hai.

Ab hum raw pushes ko $q$ se divide karke fair, size-free numbers mein badal sakte hain.

Parent note ke saare supersonic formulas mein factor $\sqrt{M_{\infty }^2-1}$ hota hai. Isse abhi mile taaki baad mein surprise na ho:

• $M_{\infty }>1$ ke liye yeh ek real, positive number hai jo $M$ ke saath badhta hai. Kyunki yeh $C_p=2\theta /\sqrt{M^2-1}$ ke denominator mein hai (aur isliye $C_D$ mein bhi), zyada supersonic speed → bada factor → surface per kam wave drag.
• $M_{\infty }=1$ pe yeh $0$ hai, aur zero se divide karna chilaata hai "theory bilkul speed of sound pe blow up hoti hai" — exactly transonic spike.
• $M_{\infty }<1$ ke liye root ke andar quantity negative hai; subsonic version $\sqrt{1-M^2}$ use karta hai. Dekho Prandtl-Glauert compressibility correction.

Related design numbers jo tum miloge: critical aur drag-divergence Mach numbers $M_{cr}$ aur $M_{dd}$ mark karte hain jahan trouble shuru hoti hai aur jahan drag spike karta hai; Area Rule aur sweep angle $\Lambda$ fixes hain.

8. Sab kuch kaise feed karta hai ek saath

Ise aise padho: pressure ek warning speed define karta hai, warning speed vs body speed $M$ define karta hai, $M$ decide karta hai ki cone (phir shock) banta hai ya nahi, shock entropy badhata hai aur stagnation pressure girata hai, aur woh loss hi wave drag hai jise $C_D$ measure karta hai.

Equipment checklist

Right side cover karo aur parent topic pe move karne se pehle har ek jawab do: