Foundations — Critical Mach number — onset of local supersonic flow

3.1.24 · D1 · Physics › Compressible Flow & Aerodynamics › Critical Mach number — onset of local supersonic flow

Yeh page assume karta hai ki tumne parent note Critical Mach number — onset of local supersonic flow ki notation bilkul nahi dekhi. Hum har symbol ko ground up se build karenge, usi order mein jis par wo ek doosre par depend karte hain.

1. Pressure $p$, density $\rho$, aur gas ki springiness $\gamma$

Kisi bhi formula se pehle, hum teen raw quantities ko naam dete hain jo hawa ke ek lump ko describe karti hain. Baad mein kuch bhi inhe use nahi kar sakta jab tak ye yahan na ho.

$\gamma$ har jagah kyun aata hai? Kyunki hawa ko compress karne se uska pressure aur temperature dono badhte hain, aur $\gamma$ un dono effects ke beech exchange rate hai. Neeche har isentropic formula actually "$\gamma$ bookkeeping" hai.

2. Speed of sound $a$ — wo yardstick jisse sab kuch measure hota hai

Ek line of dominoes ka picture socho: pehle ko giraa do aur "kuch hua" ki ek wave line mein daud jaati hai. Hawa mein, dominoes hawa ke molecules hain jo apne padosiyon se bump karte hain. Bumping speed depend karti hai gas ke kitna tightly packed aur springy hone par — exactly wahi $p$, $\rho$, $\gamma$ jo humne abhi define kiye:

$a=\sqrt{\gamma \frac{p}{\rho }}$

Yeh picture rakhna: $a$ wo maximum speed hai jis par ek gas "khud ko bata sakti hai" ki kuch aa raha hai. Jab hawa ko $a$ se tez move karne par majboor kiya jaata hai, wo aage warning nahi bhej sakti → exactly isliye "sonic jaana" special hai, aur isliye yeh poora chapter exist karta hai.

3. Mach number $M$ — sound ki units mein measured speed

Ratio kyun, "metres per second" kyun nahi? Kyunki hawa sirf apni khud ki sound speed ke relative speed ki parwah karti hai. Ek akela number $M$ capture karta hai "main sonic se kitna close hun", jo shocks ke liye sirf yehi matter karta hai.

• $M<1$ → subsonic (hawa kuch nahi outcart karti; warnings aage travel karti hain).
• $M=1$ → sonic (flow exactly apne ripples jitni tez chalti hai).
• $M>1$ → supersonic (flow apni sound ko outrun kar leti hai → shocks bante hain).

$M_{cr}$ (topic ka star) $M_{\infty }$ ki woh special value hai jahan wing par sabse bada $M_{local}$ pehli baar $1$ ko hit karta hai.

4. Subscript $\infty$ — "door, undisturbed"

Ek kilometre saamne plane ke aage khade hone ka picture karo: wahan hawa shant aur uniform hai. Wahi shant reference hai jahan hum sab comparisons anchor karte hain — wing ka local pressure dip $p_{\infty }$ ke relative measure hota hai.

5. Stagnation pressure $p_0$ — ruki hui hawa ka pressure

Wing ke exact nose-tip ka picture karo jahan flow split hoti hai aur ek streamline dead-stop ho jaati hai: woh point $p_0$ feel karta hai. Key magic fact (Isentropic Flow Relations se) yeh hai:

Woh shared constant woh bridge hai jo hume wing ke sonic point ko door ki hawa se compare karne deta hai — yeh wahi trick hai jo poori $C_{p,cr}$ derivation ke peeche hai.

$\frac{p_0}{p}={\left(1+\frac{\gamma -1}{2}{M}^2\right)}^{\frac{\gamma }{\gamma -1}}$

Ise aise padho: "ek parcel jitna tez chalega (bada $M$), uska static pressure $p$ uske frozen-still $p_0$ ke mukable utna hi kam hoga." Tez hawa = low pressure. Yeh pakad lo.

6. "Faster = lower pressure" kyun: streamtube picture

Sabse tez, sabse low-pressure spot $M_{local}=1$ tak pahunchne ka pehla candidate hai. Isliye hum surface par minimum pressure dhundhte hain.

7. Dynamic pressure $q_{\infty }$ aur pressure coefficient $C_p$

Break it down:

• Upar $p-p_{\infty }$: shant hawa se actual pressure difference. Suction peak par negative (pressure gira).
• Neeche $q_{\infty }$: pressure changes ke liye natural yardstick.

Toh $C_p$ = "flow ki apni dynamic pressure ki units mein expressed pressure dip." Negative $C_p$ = suction; zyada negative = stronger suction = faster local air. Yeh woh object hai jo hum Pressure Coefficient Cp mein track karte hain.

$q_{\infty }$ ko Mach language mein rewrite karna (step by step)

Hum ab denominator ko §2–§3 ke relations ki free-stream versions use karke rewrite karte hain. Neeche har substitution $a_{\infty }$, $p_{\infty }$, ${\rho }_{\infty }$ use karti hai — undisturbed values, local wale nahi:

Step 1 — free-stream sound speed. §2 se infinity par apply karne par, $a_{\infty }^2=\gamma p_{\infty }/{\rho }_{\infty }$, jo rearrange hoke ${\rho }_{\infty }=\gamma p_{\infty }/a_{\infty }^2$ deta hai.

Step 2 — free-stream speed in Mach. §3 se infinity par, $V_{\infty }=M_{\infty }{a}_{\infty }$, toh $V_{\infty }^2=M_{\infty }^2{a}_{\infty }^2$.

Step 3 — dono ko $q_{\infty }$ mein substitute karo: $q_{\infty }=\frac{1}{2}{\rho }_{\infty }{V}_{\infty }^2=\frac{1}{2}\left(\frac{\gamma p_{\infty }}{a_{\infty }^2}\right)\left(M_{\infty }^2{a}_{\infty }^2\right)=\frac{\gamma }{2}{p}_{\infty }{M}_{\infty }^2.$ Do $a_{\infty }^2$ cancel ho jaate hain — yahi poora point hai: sound speed gayab ho jaati hai aur sirf $M_{\infty }$ bachta hai.

Step 4 — wapas $C_p$ mein daalo: $C_p=\frac{p-p_{\infty }}{\frac{\gamma }{2}{p}_{\infty }{M}_{\infty }^2}=\frac{2}{\gamma M_{\infty }^2}\cdot \frac{p-p_{\infty }}{p_{\infty }}=\frac{2}{\gamma M_{\infty }^2}\left(\frac{p}{p_{\infty }}-1\right).$

Ratio $p/p_{\infty }$ mein group kyun karein? Kyunki §5 humein directly pressure ratios deta hai (shared $p_0$ ke through). $C_p$ ko $p/p_{\infty }$ ke terms mein likhne se hum isentropic relation seedha plug kar sakte hain — do facts bina kisi leftover $\rho$ ya $V$ ke snap together ho jaate hain.

$\boxed{C_p=\frac{2}{\gamma M_{\infty }^2}\left(\frac{p}{p_{\infty }}-1\right)}$

8. Critical pressure coefficient $C_{p,cr}$ — universal target line

Pressure ratio $p/p_{\infty }$ build karna. Sonic local point aur free stream dono ek $p_0$ share karte hain (§5). Har ek ko isentropic relation se likho aur divide karo: $\frac{p}{p_{\infty }}=\frac{p/p_0}{p_{\infty }/p_0}=\frac{p_0/p_{\infty }}{p_0/p}.$

• Free stream $M_{\infty }=M_{cr}$ par: $\frac{p_0}{p_{\infty }}={\left(1+\frac{\gamma -1}{2}{M}_{cr}^2\right)}^{\frac{\gamma }{\gamma -1}}$.
• Local point $M_{local}=1$ par: same $p_0/p$ relation mein $M=1$ substitute karo: ${\frac{p_0}{p}|}_{M=1}={\left(1+\frac{\gamma -1}{2}\cdot 1^2\right)}^{\frac{\gamma }{\gamma -1}}={\left(1+\frac{\gamma -1}{2}\right)}^{\frac{\gamma }{\gamma -1}}.$

Divide karne par, outer exponents ek bracket mein combine ho jaate hain, sonic point par pressure ratio dete hain: $\frac{p}{p_{\infty }}={\left(\frac{1+\frac{\gamma -1}{2}{M}_{cr}^2}{1+\frac{\gamma -1}{2}}\right)}^{\frac{\gamma }{\gamma -1}}.$

Ab ise §7 ke boxed $C_p$ mein $M_{\infty }=M_{cr}$ ke saath feed karo:

$C_{p,cr}=\frac{2}{\gamma M_{cr}^2}\left[{\left(\frac{1+\frac{\gamma -1}{2}{M}_{cr}^2}{1+\frac{\gamma -1}{2}}\right)}^{\frac{\gamma }{\gamma -1}}-1\right]$

Yeh sirf $\gamma$ par depend karta hai — koi airfoil shape nahi aata. Ise $(M_{\infty },C_p)$ plane mein ek fixed "finish line" curve samjho. Wing ka khud ka suction curve (§9) iske taraf daudta hai; jahan wo touch karte hain woh $M_{cr}$ hai.

9. Airfoil ka apna curve: $C_{p,0}$ aur Prandtl–Glauert

Jaise tum tez udte ho, compressibility us suction ko aur gehri kar deti hai. Sabse simple estimate (Prandtl–Glauert Compressibility Correction se) yeh hai:

$C_{p,\min }(M_{\infty })=\frac{C_{p,0}}{\sqrt{1-M_{\infty }^2}}$

Apni valid range mein, jaise $M_{\infty }$ badhta hai airfoil curve universal $C_{p,cr}$ line se milne ke liye neeche divekarti hai. Woh crossing $M_{cr}$ hai. Isse aage push karo toh tum drag divergence mein jaate ho; designers isse sweep aur area rule se bachte hain.

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