3.1.13 · D1Oblique shock waves — θ-β-M relation
Ek supersonic flow jab kisi cheez se takrakar mod lene par majboor hoti hai, toh ek tilted shock ke through compress hoti hai, aur yeh tilt sirf teen numbers se poori tarah fix hoti hai: flow kitni tez aa rahi hai, use kitna mod lena hai, aur shock kitni steep hai. Yeh samajhne ke liye hume sirf ek trick chahiye — incoming velocity ko ek aisi piece mein split karo jo shock cross karti hai (jo saara compression karti hai) aur ek aisi piece mein jo uske saath slide karti hai (jo kuch nahi karti).
Yeh page yeh assume karta hai ki aap kuch nahi jaante aur parent Oblique shock waves — θ-β-M relation ke har symbol ko ek ek karke build karta hai, jahan har ek agla symbol earn karta hai.
Definition Flow aur streamlineEk flow hawa (ya koi bhi gas) ka movement hai. Hum uski direction ek arrow se dikhate hain. Streamline woh path hai jo ek tiny dust speck trace karta jab woh saath saath ride karta — yeh "air kahan ja rahi hai" ki picture hai.
Neeche sab kuch ek incoming stream of air ke baare mein hai jo kisi obstacle se takra rahi hai. Us single arrow ko apne dimag mein rakho; hum use mod dene wale hain.
Isse pehle ki hum "supersonic" keh sakein, hume tez hone ka ek standard chahiye.
Definition Speed of sounda Speed of sound a a ≈ 340 m/s
M M = a V V a pure number hai (koi units nahi — ek speed divided by ek speed).
M < 1 subsonic — flow apne khud ke messages se slower move karti hai.M = 1 sonic .M > 1 supersonic — flow apne khud ke pressure messages se aage nikal jaati hai.
M V a ratio V / a M V Mach Angle and Mach Waves ka seed hai.
Jab M > 1
μ μ = arcsin ( M 1 )
arcsin ( 1/ M ) a V opposite side a hypotenuse V μ sin μ = hyp opposite = V a = M 1 Angle ko uski sine se recover karne ke liye hum arcsin
Parent μ sabse weak possible oblique shock kehta hai. Isliye hum ise pehle build karte hain: yeh aage aane wali har cheez ka limiting case hai.
Ab flow mein ek wedge (ek pointed ramp) rakho.
Definition Deflection angleθ physically bend hoti hai jab yeh wedge surface follow karti hai. Agar wedge surface θ θ
Picture: incoming arrow horizontal, wedge surface θ θ oncoming stream ki taraf ek compression hai (flow ek chote channel mein squeeze hoti hai).
Yeh turn smoothly nahi ho sakta — flow supersonic hai aur aage warn nahi ho sakti. Isliye sudden compression ki ek thin sheet appear hoti hai: shock . Yeh wedge ke saath nahi hoti; yeh apni khud ki steeper tilt par hoti hai.
β Incoming flow direction aur shock sheet ke beech ka angle. Hamesha β > θ
Intuition Do alag angles, inhe confuse mat karoθ air turn karti hai (wedge set karta hai). β shock lean karta hai (nature ise choose karti hai). Poora topic ek equation hai jo θ β M
Yeh master trick hai. Incoming velocity arrow V 1 shock sheet ko reference line use karke, ise do perpendicular pieces mein toddo.
Definition Normal aur tangential components
Normal component u n 1 bilkul across (perpendicular) point karta hai.Tangential component w saath point karta hai.
V 1 u n 1 = V 1 sin β , w = V 1 cos β
sin β cos β β V 1 saath wali piece us angle ki adjacent side hai → cos β across wali piece opposite side hai → sin β
Intuition Split karne ki zaroorat kyun hai?Kyunki do pieces bilkul alag behave karti hain. Normal piece compression mein jam jaati hai aur ordinary Normal Shock Waves laws follow karti hai. Tangential piece sheet ke saath untouched slide karti hai (ise change karne ke liye koi sideways force nahi hai). Ek hard collision + ek free slide = poora oblique shock. Yeh "Normal Normally, Tangent Tags Along" mnemonic hai.
Sirf normal piece "shock karti hai". Isliye effective Mach number jo normal-shock physics mein enter karta hai sirf u n 1
M 1 M 1 saari speed shock cross karti hai — lekin tangential slide kabhi nahi karti. Yeh compression ko overstate kar dega. Sirf M n 1 = M 1 sin β
Ek real shock ko M n 1 > 1 sin β > 1/ M 1 β > μ β = μ
Shock ke baad wali quantities ko subscript 2 1
M 2 θ Downstream velocity V 2 θ ( β − θ ) u n 2 = V 2 sin ( β − θ )
( β − θ ) θ taraf rotate ho gayi hai. Isliye (naye) flow aur shock ke beech angle β β − θ
Definition Tools, plain words mein
tan α = cos α sin α = adjacent opposite steepness measure karta hai. Yeh isliye appear karta hai kyunki final θ-β-M relation do "steepnesses" compare karta hai (normal vs tangential change).cot β = tan β 1 = sin β cos β cos 2 β double-angle ; yeh sin 2 β cos 2 β cos 2 β = 1 − 2 sin 2 β arcsin arctan
γ γ γ = 1.4 γ kitna . Yeh Rankine-Hugoniot Relations ke saath packaged aata hai jo kisi bhi shock govern karta hai.
ρ ρ compress karti hai, isliye ρ 2 > ρ 1 ρ 1 u n 1 = ρ 2 u n 2
split V into normal and tangential
normal Mach Mn1 = M sin beta
Ise upar se neeche padho: sound speed M M θ β M n 1 γ ρ
Right side cover karo aur reveal karne se pehle answer do.
M Flow speed V a M = V / a
M > 1 Flow apne khud ke pressure ripples se aage nikal jaati hai — supersonic.
Mach angle define karo aur uska formula batao. Ripple pile-up cone ka half-angle, μ = arcsin ( 1/ M )
θ Woh angle jitna air physically wedge se bend hoti hai.
β θ Shock sheet ka incoming flow ke vs tilt; hamesha β > θ
V 1 Normal aur tangential, dono shock sheet ke relative measure kiye.
u n 1 w V 1 β u n 1 = V 1 sin β w = V 1 cos β
sin β cos β β sin cos
Normal-shock laws mein kaunsa effective Mach number enter karta hai, aur kyun? M n 1 = M 1 sin β
β sin β > 1/ M 1 β > μ M n 1 > 1
Downstream flow shock ke saath kaunsa angle banati hai, aur kyun? ( β − θ ) θ
γ Ratio of specific heats, γ = 1.4
