Normal shock properties — M₂, P₂ - P₁, T₂ - T₁, ρ₂ - ρ₁, P₀₂ - P₀₁: Foundations

3.1.12 · D1 · Physics › Compressible Flow & Aerodynamics › Normal shock properties — M₂, P₂ - P₁, T₂ - T₁, ρ₂ - ρ₁, P₀₂

Isse pehle ki aap $M_{2}^{2} = \frac{1 + \frac{γ - 1}{2} M _{1}^{2}}{γ M _{1}^{2} - \frac{γ - 1}{2}}$ jaisi equations par trust kar sakein, aapko jaanna hoga ki har letter kya hai, woh kaunsi picture banata hai, aur yeh topic uske bina kyu nahi chal sakta. Hum ek ek symbol par jaayenge, aur koi letter tab tak use nahi karenge jab tak woh "earn" nahi ho jaata.

1. Do states: subscripts "1" aur "2"

Poora topic ek pehle/baad ki kahaani hai. Gas shock ke ek taraf se andar aati hai aur doosri taraf se baahir nikalti hai.

Isko ek darwaaze ki tarah socho. Darwaaze ke baayein taraf sab kuch "state 1" hai; daayein taraf sab kuch "state 2" hai. Shock ke baare mein koi bhi baat tab tak samajh nahi aati jab tak aap yeh tay nahi karte ki aap kis taraf khade ho.

Figure — Normal shock properties — M₂, P₂ - P₁, T₂ - T₁, ρ₂ - ρ₁, P₀₂ - P₀₁

Yeh topic iske bina kyu nahi chal sakta: parent note ki har ek equation ek ratio jaisi hai, jaise $P_{2} / P_{1}$ — baad aur pehle ka comparison. Labels ke bina, ratios ka koi matlab nahi.

2. Pressure $P$ — gas ka dhakka

Isko aise socho — kaafi saare chote molecules ek diwar par drum baja rahe hain. Zyada drumming per second, ya zyada zor se hits, matlab zyada $P$ . Neeche teal arrows gas molecules ko ek surface par bombard karte huye dikhate hain.

3. Density $ρ$ (rho) — gas kitni "bheed" mein hai

Symbol $ρ$ Greek letter "rho" hai (bolte hain "row"). Yeh letter p nahi hai — yeh sabko ek baar confuse karta hai.

Isko aise socho — ek box mein dots. Kam dots = kam density (patli hawa); bahut saare dots ek saath = zyada density (mooti hawa). Shock cross karte waqt dots aur paas aa jaate hain, isliye $ρ_{2} > ρ_{1}$ .

Yeh topic iske bina kyu nahi chal sakta: density hi "compression" ka matlab hai. Yeh famous result ki $ρ_{2} / ρ_{1}$ kabhi bhi hawa ke liye $6$ se zyada nahi ho sakta (equation 9 ki limit) — yeh ek statement hai ki dots kitni tightly pack ho sakti hain.

4. Temperature $T$ — molecules ki "jiggle energy"

Isko aise socho — blurry, tezi se vibrate karte dots (garam) versus dheeme, shant dots (thande). Supersonic flow ko squeeze karo aur dheema karo aur uski ordered forward motion random jiggle mein badal jaati hai — isliye $T_{2} > T_{1}$ . Yahi wajah hai ki hypersonic vehicles jalte hain: parent ke Example 2 mein hawa $300$ K se $1740$ K ho jaati hai.

5. Ideal gas law — $P$ , $ρ$ , $T$ ko ek saath jodhne wala glue

Yeh teeno properties independent nahi hain. Yeh ek relation se lock hain.

Isko ek see-saw ki tarah socho: volume fixed rakho aur gas ko garam karo (raise $T$ ) toh woh zyada dhakka deti hai (raise $P$ ); zyada dots pack karo (raise $ρ$ ) aur woh bhi zyada dhakka deti hai. Parent exactly isi tarah doosre doono se temperature nikalti hai: $\frac{T _{2}}{T _{1}} = \frac{P _{2} / P _{1}}{ρ _{2} / ρ _{1}}$ ( $P = ρR T$ ko rearrange karne se).

Yeh topic iske bina kyu nahi chal sakta: yeh "equation of state" hai jo system ko close karti hai — parent kehta hai ki paanch unknowns ke liye teen conservation laws plus $P = ρR T$ chahiye. Yahi woh plus hai.

6. Velocity $u$ — gas aage kitni tezi se chal rahi hai

Isko ek nadi ki dhara ki tarah socho: $u$ woh arrow hai jo paani ki travel direction aur speed dikhata hai. Shock ke paar flow slow ho jaata hai, isliye $u_{2} < u_{1}$ . Kyunki mass conserved hota hai, flow ko slow karne ka matlab hai dots ko crowd karna — isliye $ρ_{2} u_{2} = ρ_{1} u_{1}$ velocity aur density ko saath baandhta hai (parent equation 1).

7. Speed of sound $a$ — khabar kitni tezi se travel karti hai

Yeh poore chapter ka pivot hai. Sound woh tarika hai jisse gas messages pass karti hai: "rasta do, kuch aa raha hai."

Isko aise socho — ek pond mein pathar girne se failti ripples. Agar aap ripples se dheemi speed se chalte ho, toh woh aage nikal jaati hain aur aapke saamne paani ko warn karti hain. Agar aap ripples se tezi se chalte ho, toh aap apni warning se aage nikal jaate ho — saamne wala paani aapke aane se pehle kabhi nahi jaanta jab tak aap usse crash nahi karte. Woh crash hi shock hai. Neeche plum figure dono cases dikhata hai.

Yeh topic iske bina kyu nahi chal sakta: shock isliye exist karta hai kyunki flow apni awaaz se aage nikal jaata hai. $a$ ke bina, "awaaz se tez" kehne ka koi tarika nahi, aur poora idea collapse ho jaata hai.

8. Mach number $M$ — speed ko "sounds" mein measure karo

Isko ek ratio dial ki tarah socho: $M$ batata hai ki aap kitne "sound-speeds" tezi se chal rahe ho. Parent ka master result (equation 7) ek machine hai jo $M_{1} > 1$ khata hai aur hamesha $M_{2} < 1$ ugalta hai: supersonic andar, subsonic baahir.

9. Specific heats ka ratio $γ$ (gamma)

Symbol $γ$ Greek "gamma" hai. Hawa ke liye yeh $1.4$ hai.

Isko gas ki "stiffness personality" ki tarah socho: yeh decide karta hai ki pushing (pressure) kitni aasani se heating mein convert hoti hai. Aapko yahan isko derive nahi karna — bas jaano ki yeh ek constant hai jo aap saath lekar chalte ho, jo $a = γ R T$ mein aur magic number $\frac{γ + 1}{γ - 1} = 6$ mein appear karta hai jo compression ko cap karta hai.

10. Stagnation properties $T_{0}$ aur $P_{0}$ — "rok ke dekho" wali values

Socho ki moving gas ko dheere dheere, smoothly bilkul rok diya jaaye. Uski saari forward motion heat aur pressure ban jaati hai. Woh values jo tab hogi woh stagnation (ya "total") values hain, subscript $0$ ke saath.

Isko aise socho — apni hatheeli hawa ke saamne flat rakho: hatheeli ke bilkul paas wali hawa ruk gayi hai, aur woh ruki hui hawa freely streaming hawa se zyada garam aur zyada high-pressure hai. Plum-and-orange figure ek flow ko smoothly decelerate karke uske stagnation point tak le jaata dikhata hai.

11. Entropy $s$ — messiness ka meter

Isko aise socho — ek shuffle ki hui taash ki gaddi: aap order ko chaos mein free mein badal sakte ho, lekin kabhi bina mehnat ke un-shuffle nahi kar sakte. Shock neat supersonic flow ko shuffle kar deta hai, isliye $Δ s > 0$ hamesha. Woh ek fact "backward" shocks forbid karta hai aur $P_{02} / P_{01} < 1$ force karta hai. Dekho Entropy and Second Law.

Prerequisite map

Baayein taraf ka har box is page par build kiya gaya ek symbol hai; arrows dikhate hain ki woh daayein taraf ke parent topic mein kaise feed karte hain.

Equipment checklist

Daayein taraf dhak do aur khud se test karo. Agar aap har ek ka jawaab de sakte ho, toh aap parent note ke liye taiyaar ho.

Subscript

1

versus

2

ka kya matlab hai?

State 1 = upstream (shock se pehle, supersonic); state 2 = downstream (baad mein, subsonic).

Pressure

P

kya hai, words aur units mein?

Gas jo force per unit area exert karti hai; pascals mein measure hota hai,

N/m^{2}

Symbol

ρ

ka kya matlab hai aur yeh

P

se alag kaise hai?

ρ

(rho) density hai = mass per volume (bheed);

P

push per area hai — alag pictures aur alag units.

Temperature

T

ratios ke liye kelvin mein kyun hona chahiye?

T_{2} / T_{1}

jaisi ratios ke liye ek absolute zero chahiye; Celsius ka zero arbitrary hai aur galat ratios dega.

Ideal gas law batao aur yahan uska role kya hai.

P = ρR T

; yeh teeno static properties ko link karta hai aur unknowns ka system close karta hai.

u

kya hai aur shock ke paar yeh kaise change hota hai?

Bulk flow speed; yeh girta hai (

u_{2} < u_{1}

) kyunki flow subsonic ho jaata hai.

Speed of sound

a

define karo aur physically yeh kya hai.

a = γ R T

; woh speed jis par pressure ripples (warnings) gas mein travel karti hain.

Mach number

M

define karo aur

M = 1

M < 1

M > 1

ka matlab batao.

M = u / a

;

M = 1

sonic hai,

M < 1

subsonic,

M > 1

supersonic.

Poora topic

u

ki jagah

M

mein kyun likha hai?

Sirf ratio

u / a

hi decide karta hai ki flow supersonic hai ya nahi aur isliye shock banega ya nahi.

Hawa ke liye

γ

kya hai aur yeh kahaan appear karta hai?

γ = 1.4

;

a = γ R T

mein aur compression cap

\frac{γ + 1}{γ - 1} = 6

mein.

T_{0}

aur

P_{0}

physically kya hain?

Woh temperature/pressure jo gas ki hogi agar use smoothly rok diya jaaye;

T_{0}

= total energy,

P_{0}

= useful push-power.

Shock mein kaun sa stagnation quantity bachta hai aur kaun sa girta hai, aur kyun?

T_{0}

conserved rehta hai (adiabatic, energy saved);

P_{0}

girta hai kyunki shock irreversible hai (

Δ s > 0

Δ s > 0

kya forbid karta hai?

Ek "backward" shock (subsonic se supersonic); second law sirf supersonic-in, subsonic-out allow karta hai.

Parent par jaao: Normal Shock Properties. Related building blocks: Isentropic Flow Relations, Rankine–Hugoniot Relations.