Critical Mach number — onset of local supersonic flow: Visual walkthrough

3.1.24 · D2 · Physics › Compressible Flow & Aerodynamics › Critical Mach number — onset of local supersonic flow

Step 1 — Wing hawa ko squeeze karti hai, isliye hawa speed up hoti hai

KYA. Left-to-right flow kar rahi hawa ek wing se milti hai. Upar ki humped surface par streamlines (woh patli lines jinhein hawa follow karti hai) aapas mein pinch hokar paas aa jaati hain.

KYUN. Ek nadi ke baare mein socho jo ek tangi ghati se guzarne par majboor ho: utna hi paani har second guzarna chahiye, isliye jahan channel tanga hai wahan paani tez bhagta hai. Wing ke upar hawa bilkul aisa hi karti hai — curvature "channel" ko tanga kar deti hai, isliye hawa accelerate karti hai. Tez hawa ka pressure kam hota hai (yeh wing ke upar ka "suction peak" hai).

PICTURE. Red patch ko dekho — woh sabse tez, sabse kam pressure wala point hi hai jise hum poore page track karenge.

Poora khel yeh hai: kis $M_{\infty}$ par red-patch ka $M_{l oc a l}$ pehli baar $1$ ko hit karta hai? Woh $M_{\infty}$ hi critical Mach number $M_{cr}$ hai.

Step 2 — Total ("stagnation") pressure ek streamline par har jagah same rehta hai

KYA. Ek streamline lo. Bahut door upstream hawa ka static pressure $p_{\infty}$ aur Mach $M_{\infty}$ hai. Red patch par $p$ aur $M_{l oc a l}$ hai. Hum claim karte hain ki yeh ek single conserved quantity se aapas mein bande hain.

KYUN. Smooth, no-friction, no-heat flow (isentropic — dekho Isentropic Flow Relations) ke liye, streamline ke saath energy conserve hoti hai. Agar tum hawa ko dheerey se rest par le aao, woh hamesha usi "stagnation pressure" $p_{0}$ par wapas aati hai. Isliye $p_{0}$ ek fixed label hai jo poori line ke saath carry hota hai.

PICTURE. Dashed reservoir $p_{0}$ dono stations ke upar baitha hai; har station ka static pressure usi $p_{0}$ ka ek fraction hai.

Step 3 — $p_{0}$ cancel karo taaki dono stations ke beech pressure ratio mile

KYA. Red patch par $p_{0} / p$ aur door upstream $p_{0} / p_{\infty}$ likho, phir divide karo. Unknown reservoir pressure $p_{0}$ cancel ho jaata hai.

KYUN. Hum $p_{0}$ directly measure nahi kar sakte, lekin zaroorat bhi nahi hai — sirf ratio $p / p_{\infty}$ aerodynamics ke liye matter karta hai. Ek nuisance constant cancel karna is step ki poori trick hai.

PICTURE. Do "towers" of pressure jo ek hi roof $p_{0}$ share karte hain; unhe divide karne se roof mit jaati hai.

Step 4 — Pressure ko coefficient $C_{p}$ ke roop mein repackage karo

KYA. Aerodynamicists kabhi raw pressure quote nahi karte; woh dimensionless pressure coefficient $C_{p}$ quote karte hain.

KYUN. " $p = 41, 000 Pa$ " jaisa number context ke bina kuch nahi batata. $C_{p}$ measure karta hai ki pressure free-stream se kitna neeche gira, "dynamic pressure" $\frac{1}{2} ρ_{\infty} V_{\infty}^{2}$ se scale karke (woh pressure jo chalti hawa de sakti hai agar rok di jaye). Negative $C_{p}$ = suction. Dekho Pressure Coefficient Cp.

PICTURE. Ek dial $C_{p} = + 1$ (full stagnation) se neeche $0$ tak aur phir negative suction territory mein.

Step 5 — Red patch ko sonic force karo: $C_{p, cr}$ banao

KYA. $M_{l oc a l} = 1$ set karo (red patch abhi-abhi sound tak pahuncha) aur free stream ko $M_{\infty} \to M_{cr}$ rename karo (yeh woh moment hai jo hum dhundh rahe hain). Step 3 ka ratio Step 4 ke formula mein daalo.

KYUN. "Critical" ka matlab hai local flow ka pehla sonic instant. $M_{l oc a l} = 1$ substitute karna literally us condition ko algebra mein likhna hai.

PICTURE. Red patch bilkul Mach 1 par gold ho jaata hai — baad ki sab cheez ka trigger.

Step 6 — Airfoil apni KHUD ki curve laata hai

KYA. Har wing ka ek fixed incompressible minimum pressure coefficient $C_{p, 0}$ hota hai (ek negative number jo sirf shape se set hota hai). Speed badhne par, compressibility us suction ko aur gehri kar deti hai. Ek simple model hai Prandtl–Glauert.

KYUN. $C_{p, cr}$ ne bataya ki gas dynamics kya maangti hai. Ab hume chahiye ki airfoil har $M_{\infty}$ par actually kya deliver karta hai. "Demand" vs "supply" compare karne se hi hum pata laga sakte hain ki woh kahan milte hain. Dekho Prandtl–Glauert Compressibility Correction.

PICTURE. Ek halki incompressible dip jo $M_{\infty} \to 1$ ke saath gehri hoti jaati hai.

Step 7 — Intersection hi $M_{cr}$ hai

KYA. Dono curves ko ek axis par $M_{\infty}$ ke against plot karo: universal $C_{p, cr}$ (zero ki taraf uthta hua) aur airfoil $C_{p, m i n}$ (neeche girata hua). Jahan woh cross karein, wing ka suction peak abhi-abhi sonic requirement se mila hai.

KYUN. Crossing ke baayi taraf airfoil ki suction required se shallower hai — koi point abhi sonic nahi. Daayi taraf suction requirement se zyada ho gayi — supersonic pocket pehle se ban chuka hai. Crossing knife-edge hai: pehla sonic instant. Woh $M_{\infty}$ hi $M_{cr}$ hai.

PICTURE. Do pastel curves ek dot par milti hain — poori derivation ek single frame mein.

Step 8 — Edge & degenerate cases (kabhi surprised mat ho)

KYA / KYUN / PICTURE un corners ke liye jo smooth story chhupa deti hai:

Zero lift par flat plate ( $C_{p, 0} \to 0^{-}$ ): airfoil curve barely dip karti hai; woh $C_{p, cr}$ se sirf tab milti hai jab $M_{cr} \to 1$ . Ek perfectly thin, unloaded surface ka $M_{cr} \approx 1$ hota hai — kuch accelerate nahi hota, kuch jaldi sonic nahi hota.
$C_{p, cr}$ mein $M_{\infty} \to 1$ : bracket $\to 0$ aur prefactor $2/ (γ M_{cr}^{2}) \to 1.43$ , isliye $C_{p, cr} \to 0^{-}$ . Universal curve exactly Mach 1 par zero touch karti hai — sensible: Mach 1 par udte waqt sonic hone ke liye essentially koi extra suction nahi chahiye.
Bahut cambered / thick wing (bada $∣ C_{p, 0} ∣$ ): girti hui curve jaldi cross karti hai, $M_{cr}$ $0.6$ se bhi neeche aa sakta hai. Isliye transonic jets $M_{cr}$ wapas upar laane ke liye sweep aur area rule use karte hain.
$M_{cr}$ ke baad: local sonic pocket ek shock se band hota hai, drag divergence trigger hota hai — woh topic wahin se shuru hota hai jahan yeh page khatam hota hai.
Prandtl–Glauert caution: $M_{cr}$ ke paas, P–G suction ko under-predict karta hai; actual crossing (Kármán–Tsien) thodi neeche hoti hai. Ise picture ke liye use karo, final digit ke liye nahi.

Ek-picture summary

Sab kuch isme collapse hota hai: universal gas-dynamics curve (physics kya maangti hai Mach 1 reach karne ke liye) airfoil ki apni suction curve se milti hai (shape kya supply karta hai) — crossing hi $M_{cr}$ hai.

Recall Feynman retelling — poora walkthrough seedhe shabdon mein

Wing ke hump ke upar daudti hawa speed up hoti hai aur uska pressure girta hai (Step 1). Kisi bhi streamline ke saath, ek hidden "total pressure" fixed rehta hai (Step 2), isliye main fast spot par pressure ko door waale pressure se relate kar sakta hoon aur mysterious constant bas cancel ho jaata hai (Step 3). Main us pressure ko ek tidy dimensionless dip mein repackage karta hoon jise $C_{p}$ kehte hain (Step 4). Ab main poochhta hoon: us spot ko exactly sonic banana ke liye kitni dip chahiye? Local Mach ko 1 set karne se ek clean formula milta hai, $C_{p, cr}$ , jo sirf gas ki parwah karta hai, wing ki nahi (Step 5). Lekin wing ki apni suction hai jo tez udne par gehri hoti jaati hai — Prandtl–Glauert curve (Step 6). Main dono ko flight speed ke against draw karta hoon; jahan woh cross karein, wing ki suction abhi-abhi sonic demand se mili hai — aur woh flight speed critical Mach number hai (Step 7). Corner cases theek behave karte hain: flat plates Mach 1 tak sonic-free rehte hain, moti wings jaldi critical ho jaati hain, aur $M_{cr}$ ke thodi der baad shock banta hai aur drag blast hoti hai (Step 8).

Recall

$M_{cr}$ ka ek-line meaning ::: Woh flight Mach number jahan wing ka sabse kam pressure wala point pehli baar local Mach 1 reach karta hai. $C_{p, cr}$ curve ko universal kya banata hai ::: Ismein sirf $γ$ aur $M_{cr}$ hain — koi airfoil geometry nahi. Intersection $M_{cr}$ kyun deta hai ::: Yahan airfoil ki actual suction pehli baar us suction ke barabar hoti hai jo sonic hone ke liye chahiye. Patli wing ka effect ::: Chhota $∣ C_{p, 0} ∣$ crossing ko daayein shift karta hai → zyada $M_{cr}$ .