Visual walkthrough — Area-Mach number relation A - A - = f(M) — isentropic flow

3.1.6 · D2 · Physics › Compressible Flow & Aerodynamics › Area-Mach number relation A - A - = f(M) — isentropic flow

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Step 0 — Woh picture jisse hum shuru karte hain

KYA. Ek tube jiska width badalta rehta hai. Air left side ke ek bade calm reservoir se enter karti hai aur tube mein se guzarti hai. Kahin tube apne sabse narrow point tak simat jaati hai — yeh hai throat.

KYUN. Jo kuch bhi hum prove karte hain woh is streamtube ke baare mein hai, isliye pehle ise draw karna zaroori hai aur koi bhi symbol aane se pehle iske parts ko naam dena hoga.

PICTURE.

Neeche sab kuch in charon quantities ke beech ki baat hai.

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Step 1 — Mass ikatta nahi ho sakta (continuity)

KYA. Jo bhi mass of air ek doorway se har second cross karti hai, woh har doorway se har second cross karni chahiye. Hum us fixed rate ko $\dot{m}$ (kilograms per second) likhte hain: $\dot{m}=\underset{\text{kitna packed}}{\underbrace{\rho }}\underset{\text{doorway size}}{\underbrace{A}}\underset{\text{kitna fast}}{\underbrace{V}}=\text{const along the tube}$

KYUN yeh tool. Humein ek cheez chahiye jo har jagah same ho taaki do alag stations ko compare kar sakein. Conservation of mass (continuity) exactly woh anchor deta hai. Abhi kuch fancy nahi chahiye.

PICTURE. Do doorways, bilkul alag sizes, lekin air-molecules ki tadaad har second cross karti hui (red flux) identical hai.

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Step 2 — Yardstick $A^{\ast }$ invent karo

KYA. Woh ek imaginary doorway choose karo jahan air exactly sound ki speed par chal rahi ho, $M=1$. Uski area ko $A^{\ast }$, density ko ${\rho }^{\ast }$, speed ko $a^{\ast }$ (sonic speed) kaho. Kyunki $\dot{m}$ har jagah same hai, sonic-station mass flow kisi bhi station ke barabar hai: $\rho AV={\rho }^{\ast }{A}^{\ast }{a}^{\ast }$

KYUN. cm² mein raw area akele kuch nahi batata. Humein compare karne ke liye ek reference size chahiye — jaise apni height "heads" mein measure karo. Sonic doorway perfect ruler hai kyunki yeh ek isentropic streamtube ke saath constant rehta hai (same mass flow, same reservoir). Dekho Choked Flow & Maximum Mass Flow ki yeh sach mein ek true constant kyun hai.

PICTURE. Real station (left) aur imaginary sonic station $A^{\ast }$ (right, red mein), dono ek hi reservoir se fed, dono same $\dot{m}$ pass kar rahe hain.

Ab dono sides ko $\rho AV$ se divide karo... actually, jo ratio chahiye use isolate karne ke liye rearrange karo: $\boxed{\frac{A}{A^{\ast }}=\frac{{\rho }^{\ast }{a}^{\ast }}{\rho V}}$

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Step 3 — Density ratio ${\rho }^{\ast }/\rho$

KYA. Hum dono densities ko stagnation density ${\rho }_0$ (calm reservoir mein density, jahan $V=0$) ke through route karte hain. Isentropic stagnation relation kehta hai, local Mach $M$ ke liye: $\frac{{\rho }_0}{\rho }={\left(1+\frac{\gamma -1}{2}{M}^2\right)}^{\frac{1}{\gamma -1}}$

KYUN yeh tool. Hum ${\rho }^{\ast }$ ko $\rho$ se directly compare nahi kar sakte — woh alag points par hain. Lekin dono ek hi reservoir ${\rho }_0$ se jude hain. To hum ${\rho }_0$ ko common reference ke roop mein insert karte hain aur cancel kar dete hain:

=\frac{\left(1+\frac{\gamma-1}{2}\cdot 1^2\right)^{-\frac{1}{\gamma-1}}} {\left(1+\frac{\gamma-1}{2}M^{2}\right)^{-\frac{1}{\gamma-1}}}$$ Upar sirf $M=1$ set kiya hai (star ka matlab yahi hai). Term by term: - $\gamma$ — specific heats ka ratio (air ke liye $\approx 1.4$); yeh control karta hai ki gas kitni compressible hai. - Exponent $\frac{1}{\gamma-1}$ seedha isentropic law $p\propto\rho^\gamma$ se aata hai. - Numerator mein $M=1$ set karna use sonic value par freeze kar deta hai. **PICTURE.** Ek "bridge" diagram: $\rho$ aur $\rho^*$ dono shared reservoir $\rho_0$ tak climb karte hain aur cross ho jaate hain — reservoir woh common landing hai jahan dono milte hain. ![[deepdives/dd-physics-3.1.06-d2-s04.png]] --- ## Step 4 — Speed ratio $a^*/V$ **KYA.** Hamari actual speed hai $V=M\,a$, jahan $a=\sqrt{\gamma R T}$ local sound speed hai. To $a^*/V$ ek explicit $1/M$ aur ek temperature piece mein split ho jaata hai: $$\frac{a^{*}}{V}=\frac{a^{*}}{M\,a}=\frac{1}{M}\,\frac{a^{*}}{a} =\frac{1}{M}\sqrt{\frac{T^{*}}{T}}$$ **KYUN yeh tool — square root kyun?** Kyunki sound speed follow karti hai $a=\sqrt{\gamma R T}$: yeh temperature ke *square root* ke saath badhti hai. To do sound speeds ka ratio temperature ratio ka square root hota hai. Yahi ek reason hai ki $\sqrt{\;}$ appear hota hai — yeh [[Speed of Sound a = sqrt(gamma R T)|$a$ ki definition]] se inherit hua hai. Temperatures ko usi tarah reservoir $T_0$ ke through route karo jaise humne densities ke saath kiya: $$\frac{T^{*}}{T}=\frac{T^{*}/T_0}{T/T_0} =\frac{\left(1+\frac{\gamma-1}{2}\cdot 1^2\right)^{-1}}{\left(1+\frac{\gamma-1}{2}M^{2}\right)^{-1}}$$ Term by term: - $\frac{1}{M}$ — raw "kitna Mach" factor; bada $M$ matlab $V$ bada, jo $a^*/V$ ko chhota karta hai. - $\sqrt{T^*/T}$ — yeh is baat ke liye correct karta hai ki sound khud alag temperatures par alag speeds se travel karti hai. **PICTURE.** $V$ ko local sound-speed arrow $a$ ki $M$ copies ke roop mein dikhaya gaya hai; saath mein, sonic arrow $a^*$ throat temperature $T^*$ se tied hai. ![[deepdives/dd-physics-3.1.06-d2-s05.png]] --- ## Step 5 — Dono ratios multiply karo aur exponents collect karo **KYA.** Step 3 aur Step 4 ko Step 2 ki boxed line mein daalo: $$\frac{A}{A^{*}}=\underbrace{\frac{\rho^{*}}{\rho}}_{\text{Step 3}}\cdot\underbrace{\frac{a^{*}}{V}}_{\text{Step 4}} =\frac{1}{M}\left(1+\frac{\gamma-1}{2}\right)^{-\frac{1}{\gamma-1}+\left(-\frac12\right)} \left(1+\frac{\gamma-1}{2}M^{2}\right)^{\frac{1}{\gamma-1}+\frac12}$$ **KYUN.** Do constants $\left(1+\frac{\gamma-1}{2}\right)$ side by side baithe hain, to hum simply unke exponents add kar lete hain. Isi tarah do $M$-dependent brackets ke liye bhi. Exponents add karne par: $$-\frac{1}{\gamma-1}-\frac{1}{2}=-\frac{\gamma+1}{2(\gamma-1)},\qquad \frac{1}{\gamma-1}+\frac{1}{2}=+\frac{\gamma+1}{2(\gamma-1)}$$ *Negative* exponent wala constant term *positive* exponent ke saath $\frac{2}{\gamma+1}$ ke roop mein flip ho jaata hai, aur $M$-bracket ke saath combine ho jaata hai: > [!formula] Area–Mach relation, assembled > $$\frac{A}{A^{*}}=\frac{1}{M}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^{2}\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$$ > - $\frac{1}{M}$ — speed factor; chhote $M$ par dominate karta hai, ratio ko blow up karta hai. > - Bracket — temperature/density factor; bade $M$ par dominate karta hai, woh bhi blow up > karta hai. > - Dono ke beech tug-of-war ek **minimum** create karta hai. **PICTURE.** Exponent arithmetic visually laid out: do stacked exponents ek mein merge ho rahe hain. ![[deepdives/dd-physics-3.1.06-d2-s06.png]] --- ## Step 6 — Har case: finished curve padho **KYA.** $\gamma=1.4$ ke liye $A/A^*$ ko $M$ ke against plot karo aur saare regimes check karo. **KYUN.** Ek formula jise sanity-check na kar sako woh ek liability hai. Hum throat, dono limits, aur two-root feature verify karte hain — taaki reader kabhi kisi aise case se na mile jise hum skip kar gaye. **PICTURE.** ![[deepdives/dd-physics-3.1.06-d2-s07.png]] - **$M\to 0$ (dead-slow subsonic):** $1/M\to\infty$ ⇒ $A/A^*\to\infty$. Slow air ko same mass pass karne ke liye *bada* pipe chahiye — woh thin aur lazy hai. - **$M=1$ (throat):** plug in karo — bracket ban jaata hai $\left[\frac{2}{\gamma+1}\cdot\frac{\gamma+1}{2}\right]=1$ aur $1/M=1$, to $A/A^*=1$ exactly. **Minimum**. - **$M\to\infty$ (wildly supersonic):** bracket unbounded grow karta hai ⇒ $A/A^*\to\infty$ phir se. Supersonically accelerate karte rehne ke liye pipe ko *khulte rehna* padta hai ([[Area-Velocity Relation dA-A = (M^2-1) dV-V|kyunki $dA/A=(M^2-1)\,dV/V$]] sign flip karta hai). - **Two-root fact:** height $>1$ par har horizontal line U ko **do** jagah cut karti hai — ek subsonic (left), ek supersonic (right). Geometry akele nahi bata sakti kaun sa hai; back-pressure decide karta hai (dekho [[Converging-Diverging (de Laval) Nozzle]]). > [!mistake] "Koi bhi area ek Mach number deta hai." > **Kyun sahi lagta hai:** formula $M$ ka plain function lagta hai. > **Fix:** yeh *$M$ se* ek function hai, lekin one-to-one nahi. Ise *ulta* padhna (area → > Mach) U-curve ko do baar hit karta hai. Sirf ek normal shock ya chosen back-pressure hi > ambiguity resolve karta hai — dekho [[Normal Shock Waves]]. --- ## Ek-picture summary ![[deepdives/dd-physics-3.1.06-d2-s08.png]] Yeh single frame puri story carry karta hai: mass conservation do doorways ko tie karta hai, sonic doorway $A^*$ (red) yardstick hai, stagnation relations densities aur temperatures ko $M$ mein convert karte hain, aur finished U-curve kisi bhi area ratio ko Mach number mein turn karta hai (subsonic ya supersonic branch). > [!recall]- Poore walkthrough ki Feynman retelling > Socho air ek aisi pipe mein march kar rahi hai jo pinch aur swell karti hai. Rule ek: koi > gayab nahi hota — har second har doorway se utni hi air guzarti hai ($\rho A V$ constant). > Ab ek special doorway imagine karo, exactly itna bada ki air wahan sound ki speed pakde; > use $A^*$ kaho aur ruler ki tarah use karo. Kisi bhi real doorway ko us ruler se compare > karna "constant flow" ko "packing ka ratio $\times$ speeds ka ratio" mein turn karta hai. > Hum packing ratio ko rewrite karte hain calm reservoir ko middleman bana kar, aur speed > ratio split ho jaata hai plain $1/M$ mein times temperatures ka square root (square root > isliye kyunki sound speed $\sqrt{T}$ ki tarah badhti hai). Multiply karo, exponents add > karo, aur ek clean formula nikal aata hai. Use plot karo aur woh ek valley hai: bottom > Mach 1 par jahan area ruler ke barabar hai, walls infinity tak shoot karti hain dono > crawling-slow aur screaming-fast flow ke liye. Aur cheeki part — same width ek slow bheed > *ya* ek supersonic sprint fit karti hai, isliye tumhe pata hona chahiye ki tum throat ke > kis side par ho. --- ## Active recall > [!recall]- $\dot m=\rho A V$ tube ke saath constant kyun hai? > Kyunki mass conserved hai — koi air create ya destroy nahi hoti, isliye har doorway se > cross hone wale kilograms-per-second identical hone chahiye. Kaun si ek quantity do alag stations ko compare karne deti hai? ::: Mass flow $\dot m=\rho A V$, jo har jagah same hoti hai. $a^*/V$ ratio mein square root kyun appear hota hai? ::: Kyunki sound speed $a=\sqrt{\gamma R T}$ hai, isliye do sound speeds ka ratio temperature ratio ka square root hota hai. Hum dono densities (aur temperatures) ko kis common reference ke through route karte hain? ::: Stagnation reservoir values $\rho_0$, $T_0$ ke through, jo ratio lete waqt cancel ho jaate hain. $M=1$ par, bracket $\left[\frac{2}{\gamma+1}(1+\frac{\gamma-1}{2}M^2)\right]$ kitna hota hai? ::: Exactly 1, jo $A/A^*=1$ deta hai. $A/A^*\to\infty$ dono $M\to0$ aur $M\to\infty$ par kyun hota hai? ::: Chhote $M$ par $1/M$ factor blow up karta hai; bade $M$ par bracket factor blow up karta hai.