Critical Mach number — onset of local supersonic flow: Question bank

3.1.24 · D5 · Physics › Compressible Flow & Aerodynamics › Critical Mach number — onset of local supersonic flow

Shuru karne se pehle quick symbol reminders (taaki kuch bhi unexplained na rahe):

$M_{\infty}$ = free-stream Mach number — poore aircraft ki speed still air ke relative, us air mein sound ki speed se divide karke.
$M_{l oc a l}$ = skin ke ek point par air ka Mach number, jo $M_{\infty}$ se alag ho sakta hai kyunki air curves ke upar speed up ya slow ho jaati hai.
$p$ = kisi point par local static pressure; $p_{\infty}$ = bahut door upstream undisturbed pressure; $ρ_{\infty}$ = far-upstream density; $V_{\infty}$ = free-stream speed.
$C_{p}$ = pressure coefficient, exactly is tarah define kiya gaya: $C_{p} \equiv \frac{p - p _{\infty}}{\frac{1}{2} ρ _{\infty} V _{\infty}^{2}},$ matlab "local pressure undisturbed pressure se kitna neeche (negative) ya upar (positive) hai, free-stream dynamic pressure $\frac{1}{2} ρ_{\infty} V_{\infty}^{2}$ ki units mein." Dekho Pressure Coefficient Cp.
$C_{p, cr}$ = $C_{p}$ ki woh specific value jo lowest-pressure point ko reach karni chahiye taaki wahan local flow exactly sonic ho jaye ( $M_{l oc a l} = 1$ ).
$C_{p, m i n}$ = diye gaye $M_{\infty}$ par airfoil par sabse zyada negative $C_{p}$ — "suction peak." Pehla point jo sonic hoga woh yahi hai jahan ye minimum hai.
$C_{p, 0}$ = woh value jo suction peak ki incompressible flow mein hogi ( $M_{\infty} \to 0$ ) — ek fixed negative number jo sirf airfoil ki shape se tay hota hai. Ye woh "seed" hai jise compressibility correction scale karta hai.
$γ$ = ratio of specific heats (air ke liye $\approx 1.4$ ).
$Λ$ = wing sweep angle. Yahan hum leading-edge sweep use karte hain (woh angle jo leading edge flight direction ke perpendicular se banata hai), kyunki leading edge hi flow ko pehle "dekhta" hai; neeche $cos Λ$ rule isi convention ko use karta hai. Dekho Swept Wings & Transonic Design.

Definition Compressibility curve

$C_{p, 0} / 1 - M_{\infty}^{2}$ kahan se aati hai Jaise $M_{\infty}$ badhta hai, wahi wing gehri suction peak produce karta hai kyunki ab air compressible hai aur curvature ke upar ikaddi ho jaati hai. Compressible flow equations ko linearize karne par (small disturbances, thin airfoil) Prandtl–Glauert scaling milti hai: $C_{p, m i n} (M_{\infty}) = \frac{C _{p, 0}}{1 - M _{\infty}^{2}} .$ $1 - M_{\infty}^{2}$ kyun? Disturbance potential ka linearized equation Laplace's equation (incompressible) se $1 - M_{\infty}^{2}$ factor se stretch hoti equation mein badal jaata hai; us stretch ko undo karne par har pressure uske reciprocal se scale hota hai. Validity range: ye ek first-order result hai, sirf moderate subsonic $M_{\infty}$ aur thin sections par trustworthy — ye peak ko under-predict karta hai jab $M_{\infty}$ , $M_{cr}$ ke paas pahunchta hai, aur (edge cases dekho) ye $M_{\infty} \to 1$ par blows up to $- \infty$ karta hai. Dekho Prandtl–Glauert Compressibility Correction.

Intuition Shuru karne se pehle do-curve crossing ki picture banao

$M_{cr}$ do curves ko $M_{\infty}$ ke against draw karke aur unka crossing point padhke nikala jaata hai. Neeche diye figure mein black curve universal gas-dynamics requirement $C_{p, cr} (M_{\infty})$ hai — ye $M_{\infty}$ badhne par dheerey-dheerey zero ki taraf badhti hai (zyada speed par sonic reach karne ke liye kam suction chahiye). Red curve airfoil ki apni suction peak $C_{p, 0} / 1 - M_{\infty}^{2}$ hai — ye $C_{p, 0}$ se shuru hoti hai aur $M_{\infty}$ badhne par teezi se neeche girti hai. Crossing padho: woh ek $M_{\infty}$ jahan dono curves ki ek hi $C_{p}$ value hai, woh $M_{cr}$ hai, red dot aur dotted drop-line se mark kiya gaya. Uske baayein airfoil peak upar hai (requirement se kam negative) — abhi sonic nahi; uske daayein peak neeche gir chuki hai — pehle hi sonic se aage.

Figure — Critical Mach number — onset of local supersonic flow

True or false — justify karo

M_{cr}

woh free-stream Mach hai jis par poora aircraft supersonic ho jaata hai.

False. Ye woh point hai jahan flow surface par sirf ek point par sonic hoti hai — aircraft khud abhi bhi subsonic hai (

M_{\infty} < 1

), sirf ek local patch

M_{l oc a l} = 1

touch karta hai.

M_{\infty} = M_{cr}

there is already a shock wave and drag divergence on the wing.

False. Exactly

M_{cr}

par fastest point abhi-abhi

M_{l oc a l} = 1

reach karta hai; supersonic pocket aur uska terminating shock (isliye drag divergence) sirf

M_{cr}

se thoda upar jaane par dikhta hai.

The

C_{p, cr}

versus

M_{cr}

curve is different for every airfoil.

False. Ye sirf isentropic gas dynamics se aati hai aur sirf

γ

par depend karti hai — ye ek universal curve hai. Airfoil sirf apni minimum-pressure curve ke through enter karta hai.

A more negative

C_{p, cr}

means the flow is harder to drive sonic.

False.

C_{p, cr}

simply woh pressure hai jo peak ko diye gaye $M_{\infty}$ par reach karni chahiye; zyada negative ka matlab sirf hai ki us

M_{\infty}

par ek aur neechi pressure chahiye.

M_{cr}

decide karta hai dono curves ka crossing point, sirf magnitude nahi.

Airfoil ko thinner banana uska critical Mach number badhata hai.

True. Thinner section ka suction peak milder hota hai, isliye

∣ C_{p, 0} ∣

chhota hota hai; uska pressure curve universal

C_{p, cr}

curve se zyada high

M_{\infty}

par milta hai,

M_{cr}

badhata hai. Isliye fast jets ke wings thin hote hain.

Wing ko peeche sweep karna

M_{cr}

ghatata hai kyunki wing "lambi" ho jaati hai.

False, ulta effect. Sweep leading edge ke normal velocity component ko

M_{\infty} cos Λ

tak reduce karta hai (

Λ

leading-edge sweep hai); section ek slower effective flow dekhta hai, isliye local sonic reach karne ke liye true

M_{\infty}

zyada hona chahiye —

M_{cr}

badhta hai. Dekho Swept Wings & Transonic Design.

Critical pressure coefficient

M_{cr}

badhne par kam negative hota jaata hai.

True. Zyada

M_{\infty}

ka matlab hai

M_{l oc a l} = 1

reach karne ke liye kam compression chahiye, isliye required minimum

C_{p}

zero ki taraf chadh jaata hai (chhoti suction kaafi hai). Ye figure mein black curve ki gentle upward slope exactly hai.

Prandtl–Glauert kisi bhi airfoil ka exact

M_{cr}

deta hai.

False. P–G ek linearized estimate hai;

M_{cr}

ke paas ye suction peak ko under-predict karta hai, isliye ye

M_{cr}

ko over-estimate karta hai. Karman–Tsien/Laitone thode lower, zyada accurate values dete hain — P–G sirf intuition ke liye use karo.

Do airfoils jinke same incompressible

C_{p, 0}

hai unka same

M_{cr}

hota hai.

True (first order tak, P–G approximation mein). Intersection method mein

M_{cr}

sirf

C_{p, 0}

aur universal curve par depend karta hai; equal

C_{p, 0}

⇒ same crossing point. (Higher-order shape effects aur better corrections real mein tiny differences dete hain.)

Spot the error

"Kyunki air compressible hai, wing ke upar faster flow ka matlab zyada local pressure hai, isliye suction peak par

C_{p}

positive hai."

Error: faster flow ka matlab kam static pressure hai (energy pressure se speed mein jaati hai), isliye suction peak ka

C_{p}

sabse zyada negative hota hai. Ye exactly woh point hai jo pehle sonic hota hai.

"Universal

C_{p, cr}

curve aur airfoil ki

C_{p, m i n}

curve ko

M_{cr}

nikaalneके liye alag-alag Mach numbers par compare kiya jaata hai."

Error: dono ko ek hi

M_{\infty}

par padhna chahiye;

M_{cr}

woh single

M_{\infty}

hai jahan dono curves ki equal value hai (figure mein crossing). Alag

M_{\infty}

par compare karna meaningless hai.

C_{p, cr}

formula mein

M_{cr}

nikaalneके liye

M_{\infty} = 1

set karte hain."

Error: hum local point ko sonic set karte hain,

M_{l oc a l} = 1

, jabki free stream

M_{\infty} = M_{cr} < 1

par rehti hai. Dono Machs ko confuse karna classic galti hai.

"Kyunki

p_{0}

constant hai, wing par har jagah pressure ek jaisa hai."

Error: stagnation pressure

p_{0}

ek isentropic streamline ke saath constant hai, lekin static pressure

p

speed ke saath vary karta hai. Sirf

p_{0}

fixed hai;

p

suction peak par dip karta hai.

"Suction-peak curve

C_{p, 0} / 1 - M_{\infty}^{2}

zyada se zyada negative hoti rehti hai aur kabhi

C_{p, cr}

curve se cross nahi karti, isliye

M_{cr}

exist nahi karta."

Error:

C_{p, cr}

curve bhi move karti hai (

M_{\infty}

badhne par zero ki taraf), aur airfoil curve zyada teezi se girti hai, isliye ye zaroor kisi

M_{\infty} < 1

par cross karengi. Har real airfoil ka ek

M_{cr}

hota hai.

M_{cr}

se upar drag turant aur bahut zyada badh jaata hai."

Error:

M_{cr}

aur Drag Divergence Mach Number

M_{dd}

ke beech ek chhota margin hota hai; sharp drag rise

M_{dd}

par hota hai,

M_{cr}

se thoda upar, jab supersonic pocket aur uska shock strong ho jaate hain.

Why questions

Flow upper surface par pehle sonic kyun hoti hai, neeche nahi?

Upper surface ki curvature sabse zyada hai, jo streamtube ko sabse zyada constrict karti hai; continuity ke according flow wahan sabse zyada speed up hoti hai, lowest pressure aur highest

M_{l oc a l}

deti hai — isliye woh

M_{l oc a l} = 1

pehle hit karti hai.

M_{cr}

se kyun kaam lena jab plane abhi bhi subsonic hai?

Kyunki local supersonic pocket ka pehla appearance woh terminating shock seed karta hai jo drag divergence aur transonic buffet cause karta hai — woh practical "sound barrier" jo

M_{\infty} < 1

par feel hota hai.

C_{p, cr}

relation airfoil se independent kyun hai?

Ye purely isentropic flow se derive hota hai free stream aur ek sonic point ke beech (

p_{0}

constant,

M_{l oc a l} = 1

); koi geometry enter nahi karti, sirf

γ

aur

M_{cr}

Yahan sirf Bernoulli ki jagah isentropic relation kyun aati hai?

Incompressible Bernoulli in Machs par fail karta hai kyunki density change hoti hai; flow adiabatic aur reversible hai, isliye isentropic

p_{0} / p

relation compressible speeds par pressure ko Mach se correctly link karta hai.

Prefactor

2/ (γ M_{\infty}^{2})

har

C_{p}

expression mein kyun aata hai?

Ye sirf free-stream dynamic pressure

\frac{1}{2} ρ_{\infty} V_{\infty}^{2}

hai jo

\frac{γ}{2} p_{\infty} M_{\infty}^{2}

ke roop mein rewrite kiya gaya hai, isliye usse divide karne par ek raw pressure ratio dimensionless

C_{p}

mein convert ho jaata hai jo Mach-based units mein measure hota hai.

Area Rule (Transonic) kyun help karta hai jabki ye fuselage ke baare mein hai, wing ke

M_{cr}

ke baare mein nahi?

M_{cr}

ke thoda baad poora aircraft supersonic pockets grow karta hai jinke shocks add up hote hain; area rule nose se tail tak total cross-sectional area ko smooth karta hai (figure s02 dekho) taaki area mein koi abrupt jump ek strong shock na drive kare — ye transonic shock system ko weaken karta hai, drag divergence ko delay aur soften karta hai. Ye

M_{cr}

khud nahi badhata lekin uske upar jo hota hai usse tame karta hai.

Thicker airfoil ka

M_{cr}

lower kyun hota hai?

Zyada thickness ka matlab stronger curvature aur deeper suction peak (bada

∣ C_{p, 0} ∣

) hai; woh curve universal

C_{p, cr}

curve se lower

M_{\infty}

par milti hai, isliye sonic flow pehle aa jaati hai.

Edge cases

M_{\infty} \to 0

(bahut slow flight) par, two-curve picture mein kya ho raha hai?

Suction curve incompressible

C_{p, 0}

ke equal hai aur required

C_{p, cr} \to - \infty

hai; woh crossing ke paas kahin nahi hain, isliye koi local sonic flow nahi — consistent hai

M_{cr}

ke zero se kaafi upar hone ke saath.

M_{\infty} \to 1

par Prandtl–Glauert kya predict karta hai, aur ye unphysical kyun hai?

Factor

1 - M_{\infty}^{2} \to 0

, isliye

C_{p, 0} / 1 - M_{\infty}^{2} \to - \infty

— formula infinite suction predict karta hai. Woh singularity ek linearization artefact hai: flow transonic aur nonlinear ho jaati hai usse pehle hi, aur isliye P–G ko

M_{\infty} = 1

ke paas kabhi trust nahi karna chahiye (aur ek reason ye bhi hai ki ye

M_{cr}

over-estimate karta hai).

Zero angle of attack par perfectly flat plate ka

M_{cr}

kya hai?

Koi curvature nahi hai isliye koi suction peak nahi (

C_{p, 0} = 0

), isliye flow kabhi free stream se upar accelerate nahi hoti; local sonic tab hi reach hoti hai jab

M_{\infty} \to 1

, giving

M_{cr} \to 1

Agar airfoil ka incompressible suction peak

C_{p, 0}

extremely mild ho (almost zero), to

M_{cr}

kahan hai?

1

ke bahut paas — tiny suction curve fast-rising

C_{p, cr}

curve se sirf

M_{\infty} \approx 1

ke paas milti hai. Flow mein kam disturbance ⇒ local sonic baad mein onset.

Exactly

M_{\infty} = M_{cr}

par kya hota hai — kya abhi koi supersonic region hai?

Ek single sonic point hai (

M_{l oc a l} = 1

) lekin abhi koi finite supersonic pocket nahi; pocket ka zero extent hai aur grow tabhi hota hai jab

M_{\infty}

M_{cr}

se aage jaata hai.

Kya

M_{cr}

drag-divergence Mach number

M_{dd}

se zyada ho sakta hai?

Nahi — sonic flow (

M_{cr}

par) drag diverge hone ke liye shock strong hone se pehle appear honi chahiye, isliye hamesha

M_{cr} < M_{dd}

. Dekho Drag Divergence Mach Number.

γ

badhne par (alag gas), fixed

M_{cr}

par required

C_{p, cr}

zyada negative hota hai ya kam?

Ye shift karta hai kyunki isentropic exponent

γ / (γ - 1)

aur prefactor

2/ (γ M_{cr}^{2})

dono

γ

par depend karte hain; poori universal curve move karti hai, isliye kisi given airfoil ka

M_{cr}

working gas ke saath change hota hai — curve sirf fixed

γ

ke liye universal hai.

Recall Ek-line survival summary

$M_{cr}$ woh point hai jahan skin par fastest point pehli baar $M_{l oc a l} = 1$ touch karta hai jabki plane subsonic rehta hai; tum ise universal $C_{p, cr} (M_{\infty})$ curve (upar boxed formula, andar sirf $γ$ ) aur airfoil ki suction curve $C_{p, 0} / 1 - M_{\infty}^{2}$ ke crossing ke roop mein nikaalte ho — intuition ke liye acha hai lekin $M_{cr}$ ke paas under-predicting karta hai aur $M_{\infty} \to 1$ par blow up karta hai, isliye better corrections $M_{cr}$ ko thoda lower nudge karte hain. Thin + swept ⇒ higher $M_{cr}$ ; drag divergence thodi der baad aati hai.