3.1.16 · D1Prandtl-Meyer expansion waves — isentropic, supersonic turni
Jab hawa ki ek tez dhara (awaaz se bhi tez) ek aisi corner se mudti hai jo usse door jhukti hai, toh hawa ek spray ki tarah bahut patli ripples mein phail jaati hai — teez hoti jaati hai aur patli hoti jaati hai, bina kisi waste ke. Yeh poora topic bas usi spray ka bookkeeping hai: har ripple ko ek angle dena, chote-chote turns ko add karna, aur naya speed padhna.
Parent note ka koi bhi formula samajhne se pehle, tumhe ideas ka ek chhota toolbox chahiye. Neeche, parent jo bhi symbol aur word use karta hai, unhe zero se banaya gaya hai: seedha matlab → picture → topic ko yeh kyun chahiye. Upar se neeche padho; har step pichle pe khada hai.
Definition Teen gas constants pehle naam kar lete hainNeeche ka sound-speed formula teen letters use karta hai. Yahan har ek ke liye ek-line placeholder hai; §6 mein poori kahani hai:
T absolute temperature (kelvin, hamesha ≥ 0 R gas constant (air: 287 J/(kg⋅K) γ ratio of specific heats (air: 1.4
Definition Speed of sounda Speed of sound a 340 a = γ R T
M M pure number hai (koi units nahi) jo flow ki apni speed V a M = a V .
M < 1 subsonic — flow apni ripples se slower hai.M = 1 sonic — bilkul ripple-speed par.M > 1 supersonic — flow apni ripples se aage nikal jaati hai.
Figure s01 — aise padho. Amber dot flow ka source abhi hai; har cyan circle ek aisi sound ripple hai jo pehle emit hui thi, aur a t V t V > a M = 2 Mach lines par collapse ho jaata hai. Dekho ki ripples dot ke aage kabhi nahi pahunchti — woh visual hi "supersonic" ka matlab hai.
M > 1 aage tabhi spread ho sakti hai jab flow ripple se slower ho. Ek baar M > 1 Mach waves jinse yeh poora topic bana hai. No supersonic flow, no fan. Isliye M 2 − 1 M > 1
Definition Mach wave aur Mach angleμ Mach wave supersonic flow mein sabse kamzor possible disturbance hai — Figure s01 ki un seedhi safed lines mein se ek. Jo angle yeh local flow direction ke saath banati hai woh Mach angle hai, μ μ = arcsin ( M 1 ) , μ M
θ angle measure karta hai "tum kitna rotate hue ho." Parent do units use karta hai:
degrees (∘ 36 0 ∘ radians : ek poora circle 2 π ≈ 57. 3 ∘
Prandtl–Meyer function (§5 mein define ki gayi) radians mein derive hoti hai (kyunki angles ka calculus radians mein clean hota hai) lekin degrees mein quote ki jaati hai (kyunki engineers corners ko degrees mein dekhte hain). Hamesha check karo ki formula kaunsa chahta hai.
Definition Deflection angleθ wall aane wali stream se door jhukti hai — corner ki size. Flow wall ko copy karti hai, isliye θ flow direction ka bhi kitna rotate hota hai woh hai. Is topic mein θ Sign convention: θ expansion ke liye positive count karo — wall flow se door jhukti hai (convex corner), jo ν add karta hai aur flow ko speed up karta hai. Ek negative θ ek compression hogi — wall flow mein andar jhukti hai (concave corner), jo ν oblique shock banata hai. Yeh page aur parent sirf θ > 0
Common mistake Degrees ko radian formula mein daalnaKyun sahi lagta hai: ek number ek number hai.
Fix: arctan arcsin 1 0 ∘ add karo, toh mixed units ho gaye. Pehle sab kuch same unit mein convert karo — parent degrees mein kaam karta hai, isliye degrees mein add karne se pehle formula ke radian output ko degrees mein convert karo.
Is topic ka har angle ek right triangle (ek 9 0 ∘ μ
Figure s02 — kya dekhna hai. Yeh bilkul wahi triangle hai jo Figure s01 ke andar chhupa hua hai, saaf draw kiya gaya. Amber slanted side hypotenuse hai (source ka travel ∝ M ∝ 1 ∝ M 2 − 1 9 0 ∘ μ
sin μ = 1/ M a t V t sin μ = V t a t = V a = M 1 . 1/ M M 1/ M μ
Figure s03 — μ Jab tum M neeche step karte hain: Mach angle zero ki taraf shrink hota rehta hai. Woh downhill curve exactly isliye hai ki fan ki last wave (high M M
Definition Inverse trig functions aur unke rangessin , cos , tan angle lete hain aur ek ratio dete hain. Unke inverses ulti taraf jaate hain:
arcsin ( x ) kaunsa angle ka sine yeh hai?" Uska input x [ − 1 , 1 ] principal value ) [ − 2 π , 2 π ] arctan ( x ) kaunsa angle ka tangent yeh hai?" Uska input x ( − 2 π , 2 π )
Yeh principal-value ranges woh "ek jawab" hain jo calculator return karta hai. Is topic mein har argument ≥ 0 1/ M > 0 M 2 − 1 ≥ 0 [ 0 , 2 π ) μ = arcsin ( 1/ M ) woh (positive, acute) angle jiska sine 1/ M .
Definition Prandtl–Meyer functionν ( M ) ν ν ( M ) woh total angle hai jitna ek flow sonic (M = 1 M — ek running "turn score." Uska formula (parent mein derive kiya, yahan memorise nahi) hai
ν ( M ) = γ − 1 γ + 1 arctan γ + 1 γ − 1 ( M 2 − 1 ) − arctan M 2 − 1 , ν ( 1 ) = 0 domain M ≥ 1 ν M badhta hai — isliye har ν M arctan integrate karna (§8) exactly woh "which-angle" shapes produce karta hai.
Intuition Topic ko inverse kyun chahiyeTum ek ratio measure karte ho (jaise 1/ M chahte ho (wave ki slant μ ν
Definition Gas constants (poori kahani)
T absolute temperature (kelvin mein, zero se kabhi neeche nahi). Sound speed set karta hai.R specific gas constant , ek given gas ke liye fixed number (air: 287 J/(kg⋅K) γ ratio of specific heats , jo batata hai gas dabane par kitni "springy" hai. Ordinary air ke liye γ = 1.4
Yeh teeno a = γ R T 2 γ − 1 γ − 1 γ + 1 γ
Definition Static pressurep ρ T Static quantities woh hain jo ek tiny sensor jo flow ke saath drift kar raha ho read karega:
p static pressure , gas ka roz ka push per unit area.ρ density , har cubic metre mein packed mass.T static temperature , local hotness.
Yeh local values hain; fan ke through change hoti hain (teeno drop hoti hain jab flow speed up hoti hai).
Definition Stagnation ("total") quantitiesT 0 , p 0 , ρ 0 completely rok diya gaya. Uske baad jo temperature hogi woh stagnation temperature T 0 p 0 ρ 0
\frac{p_0}{p}=\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}},\qquad
\frac{\rho_0}{\rho}=\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{1}{\gamma-1}}.$$
Note karo ki har ratio ek total (subscript $0$) ko same point par ek static value ke saath pair karta hai.
Definition Upstream aur downstream: subscripts 1 aur 2Jab flow poore fan ko cross karti hai, toh hum corner se pehle ki state ko subscript 1 se label karte hain aur baad ki state ko 2 se. Toh p 1 , T 1 , M 1 p 2 , T 2 , M 2 p 2 / p 1
Intuition Expansion fan ke liye stagnation values kyun matter karte hainKyunki fan isentropic (lossless) hai, T 0 p 0 ρ 0 change nahi karte . Sirf local static T , p , ρ p 2 / p 1 p 0 p 0 ν ( M 2 ) = ν ( M 1 ) + θ
s Entropy s Second law kehta hai s
isentropic = "s
Ek shock ek single violent jump hai: woh s badhaati hai. Ek expansion fan infinitely weak Mach waves ka ek smear hai, har ek essentially zero se s Yahi wajah hai ki parent baar baar insist karta hai ki fan isentropic hai aur shock nahi.
d d M M d θ d V ek Mach wave ke aare-paar ek tiny angle se turn karti hai, phir sabhi tiny turns ko add up karti hai poora turn paane ke liye.
∫ ∫ "infinitely many infinitely small pieces add karo." Is derivation mein, ν ( M ) = ∫ 1 M ( tiny turn per bit of M ) d M d θ M = 1 ν ( M ) ν arctan answer hain.
Intuition Topic calculus se kyun nahi bach saktaEk single Mach wave flow ko sirf ek whisper se turn karti hai. Ek real corner mein hazaron aisi waves ek fan mein stacked hoti hain. "Infinitely many infinitely small pieces add karo" exactly wahi hai jo ∫ ν ( M ) arctan
Right triangle sin cos tan
Mach angle mu equals arcsin one over M
Mach number M equals V over a
Only M greater than 1 supports Mach waves
Angle in degrees and radians
Deflection angle theta with sign
arctan arcsin which angle
Prandtl Meyer function nu of M turn score
Stagnation values T0 p0 rho0
Isentropic means s constant
Differentials and integral sign
Working rule nu M2 equals nu M1 plus theta
Pressure and temperature ratios
Test karo khud ko — agar tum bina dekhe har sawal ka jawab de sako toh parent note ke liye taiyar ho.
Mach number M Flow speed V a M = V / a
Topic poora M > 1 Sirf jab flow apni ripples se aage nikal jaati hai tabhi woh slanted Mach lines (fan) mein pile hoti hain; aur
M 2 − 1 tab hi real hota hai.
Mach wave kya hai aur μ Supersonic flow mein sabse kamzor disturbance; μ μ = arcsin ( 1/ M )
Right triangle par tan μ opposite ÷ adjacent.
arcsin ( x ) "Kaunse angle ka sine x [ − π /2 , π /2 ]
Simple shabdon mein Prandtl–Meyer function ν ( M ) Woh total angle jitna flow M = 1 M M ≥ 1 ν M
sin μ = 1/ M Ripple a t V t sin μ = a / V = 1/ M
γ Ratio of specific heats (gas ki "springiness"); air ke liye γ = 1.4
Teen stagnation-to-static ratios batao. T 0 / T = 1 + 2 γ − 1 M 2 p 0 / p = ( ⋅ ) γ / ( γ − 1 ) ρ 0 / ρ = ( ⋅ ) 1/ ( γ − 1 )
Static p , ρ , T p 0 , ρ 0 , T 0 Static = local values jo ek saath-chalti sensor padhti hai; stagnation = values agar flow ko losslessly rok diya jaaye (isentropic fan mein frozen).
θ Expansion ke liye positive (wall door jhukti hai, ν
Subscripts 1 aur 2 ka kya matlab hai? State 1 upstream hai (corner se pehle), state 2 downstream hai (fan ke baad).
"Isentropic" ka kya matlab hai aur fan qualify kyun karta hai? Entropy s Δ s → 0
Derivation mein integral sign ∫ Yeh fan mein har Mach wave ke infinitesimal turns d θ ν ( M )
V = M a ln d ( ln x ) = d x / x
