Detached bow shock: Foundations — Hinglish deep dive

3.1.15 · D1 · Physics › Compressible Flow & Aerodynamics › Detached bow shock

Parent note Detached bow shock padhne se pehle, usmein use hone wala har letter aapko kuch picture karna chahiye. Neeche, har symbol ko zero se build kiya gaya hai, ek picture se anchor kiya gaya hai, aur justify kiya gaya hai — topic ko iske zaroorat kyun hai? Upar se neeche padho; har rung neeche wale par tikaa hai.

1. Speed, sound, aur Mach number $M$

Hawa ko logon ki bheed socho. Agar aap ek insaan ko dhakka do, toh "dhakka" ek insaan se doosre insaan ko ek fixed pace par pass hota hai — woh pace $a$ hai. Yeh woh speed hai jisme ek disturbance ki khabar travel kar sakti hai.

Parent $M_{1}$ likhta hai shock ke upstream (aage) flow ke Mach number ke liye. Chota 1 matlab hai "state 1 = shock se pehle"; 2 matlab hai "state 2 = shock ke baad". Isi tarah $a_{1}$ shock se pehle ki sound speed hai aur $a_{2}$ shock ke baad ki (zyada, kyunki zyada garam) sound speed hai.

Figure s01 — apni sound se aage bhaagna. Yellow dot woh body hai jo $M = 2$ par daayein taraf fly kar rahi hai. Har blue circle ek sound ripple hai jo usne pehle emit ki, bahar ki taraf speed $a$ par spread ho rahi hai. Kyunki body un ripples ke grow hone se do guna tez move karti hai, circles uske peeche pile up ho jaate hain aur unka common tangent (pink lines) ek sharp wedge banata hai — Mach wave. Dekho ki pink lines sirf isliye exist karti hain kyunki body blue circles ko outrun kar rahi hai: $M < 1$ par body har circle ke andar baithi hoti aur koi wedge nahi ban sakti thi.

Supersonic flight ke peeche wave-cone picture ke liye dekho Mach angle and Mach waves।

2. Ek vector aur uske components: $u_{n}$ aur $u_{t}$

Jab hawa ka arrow ek slanted wall (shock) se milta hai, toh hum arrow ko do perpendicular pieces mein todte hain:

$u_{n}$ — woh piece jo shock ke normal (perpendicular) hai. "Normal" yahan matlab right angle par, "sadharan" nahi.
$u_{t}$ — woh piece jo shock ke tangential (parallel) hai, uske saath slide karta hua.

Figure s02 — arrow ko split karo. Yellow line shock wall hai. White arrow incoming velocity $u$ hai. Hum ise pink piece $u_{n}$ (seedha wall ke aar-paar) aur blue piece $u_{t}$ (wall ke saath slide karta hua) mein todte hain. Dekho ki blue piece yellow wall ke flat along hai jabki pink piece usse aar-paar stabbing karti hai — sirf pink piece shock se slow hoti hai; blue wali doosri taraf bhi unchanged rehti hai.

3. Trig tools: $sin$ , $cos$ , $tan$ , $cot$

Parent ka key formula inse bhara hua hai. Har ek ek right triangle par ek specific sawaal ka jawaab deta hai. Ek corner par ek angle wala right triangle socho: usse opposite side opposite hai, uske baad ki side (long slanted wali nahi) adjacent hai, aur long slanted side hypotenuse hai. Hum inhe §4 ke angles se pehle banate hain kyunki woh angles exactly inhi tools se measure hote hain.

4. Angles: $θ$ , $β$ , aur $μ$

Teen angles poore topic mein chalte hain. Inhe straight rakkho. Har ek §3 ke trig tools se measure hota hai.

$β$ ki range note karo: yeh sabse weak shock ( $β = μ$ ) aur sabse strong ( $β = 90°$ , ek head-on normal shock) ke beech rehti hai.

Figure s03 — teen angles. Dashed white line incoming flow direction hai. Yellow line shock hai, $β$ (yellow arc) se us flow se tilted. Pink arrow shock ke baad ka flow hai, $θ$ (pink arc) se bend hua. Dono arcs compare karo: $β$ (yellow) hamesha $θ$ (pink) se bada hota hai — shock wall flow ke bend hone se zyada tilt karti hai. Blue note yaad dilata hai ki $μ$ sabse chota $β$ hai jo wall kabhi le sakti hai.

$θ$ aur $β$ saath mein kaise dance karte hain iske liye dekho Oblique shock waves, aur weak/strong split ke liye dekho Maximum deflection angle and weak/strong shock solutions।

5. Fluid state symbols: $ρ$ , $γ$

Density ratio $ρ_{2} / ρ_{1}$ shock ki physical heart hai: mass conserved hai, toh agar flow slow hoke crowd up hoti hai, toh woh ratio aapko exactly batata hai kitna.

6. Subscripts padhna aur " $n$ " tag

Labels ko saath rakkho taaki parent ki dense notation plain English jaisi padhe:

Symbol	Padho
$M_{1}$	Mach number shock se pehle
$M_{2}$	Mach number shock ke baad
$a_{1}, a_{2}$	sound speed shock se pehle / baad
$M_{n 1}$	shock se pehle Normal Mach number $= M_{1} sin β$
$M_{n 2}$	shock ke baad Normal Mach number
$ρ_{1}, ρ_{2}$	density pehle / baad
$u_{n 1}, u_{n 2}$	normal velocity pehle / baad
$u_{t}$	tangential velocity (dono taraf same!)

$M_{n 1}$ aur $M_{n 2}$ dono normal Mach numbers hain — perpendicular velocity component divided by local sound speed. Precisely: $M_{n 1} = u_{n 1} / a_{1}$ (§1 ki cool upstream sound speed $a_{1}$ use karte hue) aur $M_{n 2} = u_{n 2} / a_{2}$ (hotter downstream sound speed $a_{2}$ use karte hue). Yeh hume oblique shock ke perpendicular slice par seedha-on normal-shock math reuse karne dete hain.

7. Stand-off distance $Δ$ aur nose radius $R_{n}$

Sab kuch topic mein kaise jaata hai

Neeche ka map bottom to top padha jaata hai, jaise ek tower banana. Sound speed hume Mach number define karne deti hai. Trig hume teen angles measure karne aur unhe velocity components ke saath θ–β–M relation mein combine karne deta hai. Us relation ka ek peak hota hai — $θ_{ma x}$ — aur body ka demanded turn us peak ko clear karta hai ya nahi yeh decide karta hai ki shock attached rehti hai ya bow shock mein detach ho jaati hai. Stand-off scaling phir batati hai ki detached shock kitni door aage khadi hai.

Equipment checklist

M = 2

physically kya matlab hai?

Flow sound se do guna tez move karti hai; hawa ko koi pressure warning nahi milti jab tak body almost na aa jaaye.

Ek shock par

u

u_{n}

aur

u_{t}

mein kyun split karein?

Shock sirf apne normal ke along push karta hai, toh sirf

u_{n}

change karta hai;

u_{t}

bina kisi change ke guzar jaata hai.

θ

aur

β

mein kya fark hai?

θ

= woh angle jitna flow bend hota hai (jo body maangti hai);

β

= shock wall ka tilt (jo shock offer karta hai).

θ_{ma x}

kya hai?

Ek given

M_{1}

par ek attached oblique shock jo sabse bada deflection produce kar sakti hai — θ vs β curve ka peak; isse zyada demand karo aur shock detach ho jaati hai.

μ = arcsin (1/ M_{1})

mein

arcsin

kya kar raha hai, aur yeh kab valid hai?

Yeh sine ko undo karta hai — woh angle return karta hai jiska sine

1/ M_{1}

hai; sirf

M_{1} \geq 1

ke liye valid hai taaki input

\leq 1

rahe.

tan ϕ

aur

cot ϕ

ko triangle sides ke ratios ke roop mein likho.

tan ϕ =

opposite/adjacent;

cot ϕ =

adjacent/opposite.

tan θ

value do shock angles kyun de sakti hai?

tan

repeat karta hai, toh ek single tangent value ek se zyada angle par map hoti hai — weak aur strong solutions.

Weak solution kaun sa hai aur strong kaun sa?

Weak = chota

β

(

μ

ke paas, flow usually supersonic rehti hai); strong = bada

β

(

90°

ke paas, flow subsonic ho jaati hai).

γ = 1.4

kya describe karta hai?

Air ki springiness (ratio of specific heats), jo yeh set karta hai ki compress hone par woh kitni garam hoti hai.

M_{n 1}

aur

M_{n 2}

poori tarah se bolo.

M_{n 1} = u_{n 1} / a_{1} = M_{1} sin β

(pehle normal Mach, upstream sound speed use karte hue);

M_{n 2} = u_{n 2} / a_{2}

(baad mein, downstream sound speed use karte hue).

M_{n 1} = M_{1}

kab hota hai?

Jab

β = 90°

ho, yaani ek head-on normal shock (bow-shock centreline).

Δ

kya hai,

R_{n}

se kyun divide karein, aur correlation ki kya limits hain?

Shock–nose gap; nose radius se divide karne par yeh scale-free ban jaata hai;

0.143 e^{3.24/ M_{1}^{2}}

fit ek sphere ke liye air mein roughly

M_{1} ≳ 2

par hold karta hai, arbitrary shapes/gases ke liye nahi.

M_{1} \to \infty

par

Δ/ R_{n}

kya approach karta hai?

Ek chota finite value, around

0.143

(shock body se chipak jaati hai).

Recall Padhne ke liye taiyyar check

Agar aap bina dekkhe upar ke terahon ka jawaab de sakte hain, toh Detached bow shock kholein — wahan har symbol ab plain words jaisa padhega.