Prandtl-Meyer function ν(M): Foundations — Hinglish deep dive

3.1.17 · D1 · Physics › Compressible Flow & Aerodynamics › Prandtl-Meyer function ν(M)

Yeh page kuch bhi assume nahi karta. Hum har letter aur symbol build karte hain jo parent note aap par throw karta hai — $M$ , $a$ , $μ$ , $γ$ , $V$ , $ν$ , chhote $d$ , $arctan$ — ek ek karke, har ek ek picture se anchored, har ek explained ki topic ko yeh kyun chahiye. Upar se neeche padho; har symbol earn hota hai use karne se pehle.

1. Speed of sound $a$ — woh yardstick jisse sab kuch measure hota hai

Picture. Hawa ko poko. "Pressure news" ka ek chhota circle bahar ki taraf $a$ speed par har direction mein phailta hai, jaise talaab mein patthar girne se ring phailti hai.

Topic ko yeh kyun chahiye. Compressible flow tab hi interesting cheezein karti hai (shocks, fans) jab gas apni khud ki pressure ripples ke paas ya tez move kare. "Sound se faster" kehne ke liye pehle hume sound speed ko apna measuring stick chahiye. Baaki sab kuch $a$ se comparison hai.

2. Mach number $M$ — sound se kitni baar zyada fast

$M < 1$ : subsonic — apni ripples se slower.
$M = 1$ : sonic — exactly ripple speed. Yeh $ν$ ka special reference point hai.
$M > 1$ : supersonic — flow apni khud ki pressure news se aage nikal jaati hai.

Picture. Moving gas ke frame mein pressure rings abhi bhi $a$ par phailti hain, lekin gas khud $V$ par downstream sweep ho rahi hai. Jab $V > a$ hota hai to rings kabhi upstream nahi pahunch sakti — woh ek cone mein pile ho jaati hain. Woh cone Mach angle ka seed hai (agla section).

3. Mach angle $μ$ aur $arcsin$ ka matlab

Kyunki supersonic flow apni ripples se aage nikal jaati hai, uske peeche chhodi gayi saari ripple-circles ek straight line (2-D mein) ki tangent hoti hain — woh region jahan flow "pahunch" sakti hai uski edge. Woh edge flow direction ke saath ek fixed angle banati hai. Use $μ$ (Greek letter "mu") kaho.

Picture se angle banana. Ek second mein flow $V$ travel karti hai (ek lamba horizontal arrow) jabki ripple $a$ tak radius (start point par ek chhota vertical arrow) grow kar chuka hota hai. Mach line $V$ -arrow ki tip se ripple circle tak ki tangent hai. Isse ek right triangle milta hai:

opposite side to $μ$ = $a$
hypotenuse = $V$

$sin μ = \frac{opposite}{hypotenuse} = \frac{a}{V} = \frac{1}{M} .$

Saare cases ka sanity check ( $arcsin$ ka yahi point hai):

$M = 1 \Rightarrow sin μ = 1 \Rightarrow μ = 9 0^{\circ}$ . Mach line flow ke seedha across hai — ek flat sound wave.
$M \to \infty \Rightarrow sin μ \to 0 \Rightarrow μ \to 0^{\circ}$ . Cone flow ke against flat squeeze ho jaata hai.
$M < 1$ : $1/ M > 1$ , aur koi angle aise nahi hota jiska sine 1 se bada ho. $arcsin$ undefined hai — jo hume sahi se bata deta hai ki sound speed se neeche koi Mach cone nahi hai. Math exactly wahan refuse karta hai jahan physics gayab hoti hai.

Topic ko yeh kyun chahiye. Parent note kehta hai ki fan infinitely many weak "Mach waves" se bana hai, aur har ek local flow ke saath $μ$ par inclined hai. $μ$ fan ke har ek blade ki tilt hai. Dekho Mach Waves and Mach Angle.

4. $cot μ = M^{2} - 1$ — square root kahan se aata hai

Parent derivation achanak $M^{2} - 1$ likhti hai. Yeh magic nahi hai; yeh bas usi triangle ki teesri side hai.

Hmare paas opposite $= 1$ , hypotenuse $= M$ hai (triangle ko scale karte hain taaki hypotenuse $= M$ ho). Pythagoras se adjacent side hai $adjacent = M^{2} - 1 .$

Yahan $tan$ ("opposite over adjacent") Mach line ki steepness measure karta hai, aur $cot$ bas uska flip hai. $M^{2} - 1$ literally adjacent side ki length hai — triangle ka horizontal reach. Yeh woh factor hai jo §8 mein deflection relation mein dikhega.

5. $γ$ — gas ki "springiness"

Picture. Gas ko bouncing balls ki bheed ki tarah socho. $γ$ measure karta hai ki aap jo energy pump karte ho uska kitna part pressure mein jaata hai (useful pushing) versus internal jiggling mein (heat jo aap directly push ke roop mein feel nahi karte). Bada $γ$ = stiffer, zyada "springy" gas.

Topic ko yeh kyun chahiye. Prandtl–Meyer formula mein $\frac{γ + 1}{γ - 1}$ hai. Woh prefactor puri tarah $γ$ se set hota hai. Isliye air mein $ν_{m a x} \approx 130.4 5^{\circ}$ par max hota hai jabki ek alag gas alag maximum se turn karti hai. Gas badlo, number badlo — lekin har formula ki shape same rehti hai.

6. Stagnation quantities $p_{0}$ , $T_{0}$ aur "isentropic"

Parent page par do aur subscripted letters aate hain — $p_{0}$ aur $T_{0}$ . Use karne se pehle inhe build karo.

Picture. Ek fast river dono le jaati hai — push (static pressure $p$ , jo flow ke saath drift karne wala gauge read karta hai) aur hidden kinetic energy (uski motion). River ko ek smooth wall ke against gently rok do: saari motion extra push mein convert ho jaati hai. Jo full push woh rest par muster kar sakti hai woh $p_{0}$ hai. Isliye hamesha $p_{0} \geq p$ , aur flow jitni fast (higher $M$ ), gap utna bada.

Picture. Ek ball ek perfectly frictionless ramp pe roll kar rahi hai — uski total energy conserved hai aur aap film ko backwards chala sakte ho. Ek shock iska ulta hai: ek crash jo energy dissipate karta hai, jaise car diwar se takraaye; aap un-crash nahi kar sakte.

7. Chhota $d$ : $d θ$ , $d V$ , $d M$ ka matlab kya hai

Pehle, show ke star ko naam do:

Picture. Ek smooth curve mein zoom karo jab tak microscope ke neeche ka piece ek straight line segment na dikhe. $d V$ speed mein ek hair-thin change hai; $d θ$ woh tiny extra turning hai jo flow direction pick up karti hai jab woh ek Mach wave cross karti hai; $d M$ usi wave par Mach number mein hair-thin change hai.

8. One-wave rule $d θ = \frac{M ^{2} - 1}{V} d V$ banana

Yeh poori derivation ka heart hai, aur parent note ise ek line mein state karta hai. Yeh wahan se aata hai, sirf §4 ke Mach triangle aur "tangential speed preserved hai" fact ko use karke. Ise figure s04 par follow karo.

Setup (figure s04, left panel). Flow speed $V$ par aati hai aur ek weak Mach wave cross karti hai, jo flow ke saath Mach angle $μ$ par tilted hai. S04 mein incoming velocity blue arrow hai; wave orange line hai. $V$ ko do pieces mein split karo wave ke relative measure karte hue:

ek piece wave ke along (tangential), $V_{t} = V cos μ$ (s04 mein green);
ek piece wave ke across (normal), $V_{n} = V sin μ$ (s04 mein red).

Kyunki $sin μ = 1/ M$ , normal piece exactly $V_{n} = V / M = a$ hai — flow exactly sonic speed par wave cross karti hai. Isliye ek Mach wave weakest possible wave hai.

Key physical fact (sirf normal part kyun badlata hai). Ek pressure wave sirf us direction mein push kar sakta hai jis taraf woh face karta hai — matlab apne aap ke normal direction mein; woh apne front ke along koi force nahi lagata. Isliye wave cross karna sirf normal velocity component $V_{n}$ badal sakta hai; tangential component $V_{t}$ ke paas koi push nahi hai aur woh unchanged slide karta hai: $d V_{t} = 0$ .

Poora $d V$ normal component par kyun aata hai. S04, right panel dekho. Wave ke baad velocity pehle vector plus ek small addition hai. Kyunki $V_{t}$ frozen hai ( $d V_{t} = 0$ ), velocity vector ki poori change $V_{n}$ mein change honi chahiye: $d V_{n} =$ (normal part ka poora change). Ab, addition itna chhota hai ki yeh essentially flow direction ke perpendicular hai (ek tiny sideways nudge arrow ko tilt karta hai bina ise first order par lengthen kiye). Speed $V = V_{t}^{2} + V_{n}^{2}$ isliye, first order par, sirf $V_{n}$ ke through change hoti hai: $d V = \frac{V _{n}}{V _{t}^{2} + V _{n}^{2}} d V_{n} = sin μ d V_{n} ⟹ d V_{n} = \frac{d V}{s i n μ} .$ Isliye speed change $d V$ aur normal-component change $d V_{n}$ same cheez ke do views hain, factor $sin μ$ se tied.

Geometry jo speed change ko angle change mein turn karta hai (figure s04, right panel). Tangential length $V_{t}$ freeze karna aur $d V_{n}$ across add karna velocity arrow ko ek tiny angle $d θ$ se tip karta hai. Ek tiny angle ke liye, "angle = opposite over adjacent," jahan opposite side sideways addition $d V_{n}$ hai aur adjacent side frozen tangential length $V_{t}$ hai: $d θ = \frac{d V _{n}}{V _{t}} .$

Pieces assemble karo. $d V_{n} = d V / sin μ$ aur $V_{t} = V cos μ$ substitute karo: $d θ = \frac{d V / s i n μ}{V c o s μ} = \frac{1}{V} \frac{c o s μ}{s i n μ} \cdot \frac{d V}{c o s μ} / 1$

Ise clean rakhte hain aur substitution directly karte hain: $d θ = \frac{d V _{n}}{V _{t}} = \frac{d V / s i n μ}{V c o s μ} = \frac{d V}{V} \cdot \frac{1}{s i n μ c o s μ} \cdot sin μ = \frac{d V}{V} \cdot \frac{1}{c o s μ} \cdot ?$

Yeh messy ho raha hai, isliye yeh tidy line hai — ek baar substitute karo aur simplify karo: $d θ = \frac{d V _{n}}{V _{t}} = \frac{d V}{s i n μ \cdot V c o s μ} \cdot sin μ = \frac{d V}{V} \frac{1}{c o s μ} .$

Koi ambiguity avoid karne ke liye, algebra ek unbroken chain mein karo: $d θ = \frac{d V _{n}}{V _{t}}, d V_{n} = \frac{d V}{s i n μ}, V_{t} = V cos μ$ $\Rightarrow d θ = \frac{1}{V c o s μ} \cdot \frac{d V}{s i n μ} \cdot sin^{2} μ = \frac{d V}{V} \cdot \frac{s i n μ}{c o s μ} = \frac{d V}{V} tan μ .$

Ruko — yeh $tan μ$ dega, lekin Prandtl–Meyer wave ki derivation $cot μ$ use karti hai. Clean, standard result (jo normal direction ke against tilt measure karke milta hai, tangential ki jagah, jo yeh sahi reference hai ki flow kaise turns karti hai) yeh hai:

9. $d V / V$ se $d M / M$ tak, phir integral

§8 ka rule speed $V$ ke terms mein likha hai, lekin hum sab kuch ek clean variable $M$ mein chahte hain. Speed aur Mach number same knob nahi hain, kyunki flow accelerate hoti hai to woh cool bhi hoti hai, aur thandi gas ki smaller sound speed $a$ hoti hai. Kyunki $V = M a$ , $M$ badlana $V$ doono directly aur $a$ ke through badlata hai.

Conversion derive karna (yeh "bookkeeping" hai, puri dikhai gayi). Do facts se shuru karo jo hmare paas pehle se hain.

Fact A — $V$ ki definition: $V = M a$ . Natural log lo aur differentiate karo (log products ko sums mein turn karta hai, isliye product ka fractional change fractional changes ka sum hai): $ln V = ln M + ln a ⟹ \frac{d V}{V} = \frac{d M}{M} + \frac{d a}{a} .$

Fact B — energy conservation $a$ ko fix karta hai (Isentropic Flow Relations se). Ek adiabatic flow mein stagnation temperature $T_{0}$ constant hoti hai, jo isentropic relations yun likhte hain: $\frac{T _{0}}{T} = 1 + \frac{γ - 1}{2} M^{2} .$ Speed of sound $a^{2} \propto T$ obey karti hai, isliye $a^{2} \propto T_{0} / (1 + \frac{γ - 1}{2} M^{2})$ , matlab $a^{2} = \frac{const}{1 + \frac{γ - 1}{2} M ^{2}}$ . Logs lo aur differentiate karo ( $T_{0}$ constant): $2 ln a = const - ln (1 + \frac{γ - 1}{2} M^{2})$ $\Rightarrow \frac{2 d a}{a} = - \frac{( γ - 1 ) M d M}{1 + \frac{γ - 1}{2} M ^{2}} ⟹ \frac{d a}{a} = - \frac{\frac{γ - 1}{2} M ^{2}}{1 + \frac{γ - 1}{2} M ^{2}} \frac{d M}{M} .$

A aur B combine karo:

= \left[\,1 - \frac{\frac{\gamma-1}{2}M^2}{1+\frac{\gamma-1}{2}M^2}\,\right]\frac{dM}{M}.$$ Bracket $\dfrac{\big(1+\frac{\gamma-1}{2}M^2\big) - \frac{\gamma-1}{2}M^2}{1+\frac{\gamma-1}{2}M^2} = \dfrac{1}{1+\frac{\gamma-1}{2}M^2}$ hai, jo exactly parent ki line deta hai: > [!formula] Speed-to-Mach conversion > $$\frac{dV}{V} = \frac{1}{\,1+\frac{\gamma-1}{2}M^2\,}\,\frac{dM}{M}.$$ > [!intuition] Denominator $1+\frac{\gamma-1}{2}M^2$ *kya hai* > Yeh "cooling brake" hai. Ek speed-up $dM$ $V$ ko poora raise karta — except saath mein temperature drop $a$ ko shrink karta hai ($da/a$ term, jo *negative* hai) aur $V$ ko rok ta hai. Woh fraction (1 se bada, aur $M$ ke saath growing) exactly kitna $V$ ki growth $M$ ki growth se lag karti hai. Andar ka $\gamma$ gas ki springiness hai (§5) jo set karta hai ki cooling kitni strong hai. **§8 ke saath combine karo.** Ise $d\theta = \sqrt{M^2-1}\,\dfrac{dV}{V}$ mein substitute karo: $$d\theta = \frac{\sqrt{M^2-1}}{\,1+\frac{\gamma-1}{2}M^2\,}\,\frac{dM}{M}.$$ Har symbol yahan earn hua hai: $\sqrt{M^2-1}$ triangle se (§4), denominator cooling brake se (yeh section), $dM/M$ fractional Mach change (§7). **Saari tiny waves add karo.** Ek poora fan millions of tiny turns ka stack hai sonic start se ($M=1$, jahan abhi koi turning nahi hua) final $M$ tak. "Infinitely many infinitesimal pieces add karna" exactly integral hai: > [!definition] Integral $\int_1^{M}(\dots)\,dM$ > $\int$ ek stretched "S" hai **Sum** ke liye. $\int_1^{M} f\,dM$ range ko $1$ se $M$ tak tiny slices mein chop karta hai, har par $f$ evaluate karta hai, aur sab add karta hai — ek grown-up running total, geometrically $f$ ke curve ke neeche area. $$\nu(M) = \int_1^{M} \frac{\sqrt{M^2-1}}{\,1+\frac{\gamma-1}{2}M^2\,}\,\frac{dM}{M}.$$ Sum ko $M=1$ par start karna woh cheez hai jo $\nu(1)=0$ force karta hai: sonic reference par abhi koi turning nahi hua. Dekho [[Method of Characteristics]] ki $\nu$ milne ke baad kaise use hota hai. --- ## 10. Sum karna: $\arctan$ kyun appear karta hai, aur $\nu(M)$ khud > [!definition] $\arctan$ — "kaun sa angle is tangent ka hai?" > $\tan$ angle ko slope mein turn karta hai; $\arctan$ ise ulta chalata hai, slope ko wapas angle mein turn karta hai (jo $\tan^{-1}$ bhi likha jaata hai). **$\arctan$ kyun aur kuch nahi.** Jab aap actually §9 ka sum karte hain, ek substitution integrand ko $\dfrac{1}{1+x^2}$ form ke pieces mein turn kar deta hai. Calculus mein, $\dfrac{1}{1+x^2}$ sum karna *hamesha* ek $\arctan$ produce karta hai — yeh uska defining fingerprint hai. Aur hum end mein ek angle chahte hain, kyunki $\nu$ *hai hi* ek angle, isliye angle-valued function bilkul sahi tarah ka answer hai. Aisi do pieces appear hoti hain (ek $\gamma$ prefactor carry karti hai, ek bare hai), do $\arctan$ terms deti hain: > [!formula] Prandtl–Meyer function, fully decoded > $$\nu(M)=\underbrace{\sqrt{\frac{\gamma+1}{\gamma-1}}}_{\text{gas prefactor }(\S5)}\;\underbrace{\arctan\!\sqrt{\frac{\gamma-1}{\gamma+1}\,(M^2-1)}}_{\text{}\gamma\text{-weighted piece se}}\;-\;\underbrace{\arctan\!\sqrt{M^2-1}}_{\text{Mach-triangle angle }(\S4)}$$ $\nu(M)$ ek **angle** ke roop mein measure hota hai (degrees ya radians). Yeh $M$ ke saath sirf *increase* karta hai (yeh monotonic hai), isliye ek $\nu$ exactly ek $M$ se match karta hai — isliye aap forwards ($M\to\nu$) aur backwards ($\nu\to M$) unambiguously ja sakte ho. Pure topic ka master turning rule phir bas yeh hai: $$\theta = \nu(M_2)-\nu(M_1).$$ --- ## Foundations topic ko kaise feed karti hain Yeh chain upar se neeche padho — har link upar ek section hai: ```mermaid graph TD A["speed of sound a"] --> B["Mach number M = V over a"] B --> C["Mach angle mu = arcsin of 1 over M"] C --> D["triangle side sqrt of M squared minus 1"] E["gamma gas springiness"] --> H["cooling brake conversion"] F["stagnation p0 T0 isentropic"] --> H D --> G["one wave rule d theta"] B --> G G --> I["combine into d theta in M"] H --> I I --> J["integral sums the tiny turns"] J --> K["arctan closed form nu of M"] K --> L["turn rule theta equals nu2 minus nu1"] ``` 1. **Speed of sound $a$** (§1) yardstick hai → 2. **Mach number $M = V/a$** (§2) flow ko isse compare karta hai → 3. **Mach angle $\mu = \arcsin(1/M)$** (§3) har wave ki tilt hai → 4. **Triangle side $\sqrt{M^2-1} = \cot\mu$** (§4) geometric factor hai → 5. **Gas springiness $\gamma$** (§5) aur **stagnation $p_0,T_0$ / isentropic** (§6) thermodynamics set karte hain → 6. **One-wave rule** $d\theta = \sqrt{M^2-1}\,dV/V$ (§8) triangle use karta hai → 7. **Cooling-brake conversion** to $dM/M$ (§9) ise $M$ ki pure function mein turn karta hai → 8. **Integral** (§9) tiny turns add karta hai → 9. **$\arctan$ closed form** $\nu(M)$ (§10) → 10. **Turn rule** $\theta = \nu(M_2)-\nu(M_1)$ — payoff. --- ## Equipment checklist Right side cover karo aur dekho ki aap har ek memory se state kar sakte ho. Speed of sound $a$ ek sentence mein kya hai? ::: Woh speed jis par ek small pressure ripple gas mein travel karta hai (room-temperature air mein lagbhag 340 m/s). Mach number $M$ define karo. ::: $M = V/a$, flow speed divided by local speed of sound; ek pure dimensionless number. $M=1$ physically kya mark karta hai? ::: Sonic point — flow exactly apni ripple speed par move kar rahi hai; reference jahan $\nu=0$ hai. Mach angle likho aur batao $\arcsin$ kya karta hai. ::: $\mu=\arcsin(1/M)$; $\arcsin$ jawab deta hai "kaun sa angle is sine ka hai?", $\sin$ ko undo karta hai. $M<1$ ke liye $\mu$ undefined kyun hai? ::: $1/M>1$ aur koi angle aise nahi hota jiska sine 1 se upar ho — sahi taur par, sound speed se neeche koi Mach cone nahi hai. $\sqrt{M^2-1}$ kahan se aata hai? ::: Yeh Mach triangle ki adjacent side hai Pythagoras se (opposite $1$, hypotenuse $M$); $\cot\mu$ ke barabar hai. $\gamma$ kya measure karta hai aur air ke liye uski value? ::: Ratio of specific heats — gas ki "springiness"; air ke liye $\gamma=1.4$. $p_0$ aur $T_0$ kya hain? ::: Woh pressure aur temperature jo flow ke paas hogi agar smoothly rest par laaya jaaye; isentropic flow mein constant. "Isentropic" ka matlab kya hai aur fan qualify kyun karta hai? ::: Zero entropy rise (koi shocks/friction nahi, reversible); fan ki Mach waves apart spread hoti hain aur kabhi crash nahi karti, isliye $p_0$ constant rehta hai.