Similarity — geometric, kinematic, dynamic; Reynolds similarity: Visual walkthrough

2.2.27 · D2 · Physics › Fluid Mechanics › Similarity — geometric, kinematic, dynamic; Reynolds similar

Hum sab kuch zero se banate hain. Tumhe sirf itna pehle se pata hona chahiye: force kya hai (ek push ya pull), speed ka matlab kya hai (distance ÷ time), aur cheezein size rakhti hain (ek length).

Step 1 — Ek fluid particle se milte hain

Yahan se kyun shuru karein. Similarity ultimately is cube pe forces ke baare mein hai. Agar do alag flows apne chhote cubes ko same balance of ways mein push karein, toh cubes same paths trace karte hain, aur dono flows ek doosre ki scaled copies lagte hain. Isliye flows compare karne se pehle hum cube pe har force ka naam lete hain.

PICTURE. Cube ki ek size hai jise hum $L$ kehte hain (ek length, metres mein), yeh speed $V$ se move karta hai (metres per second), yeh density $ρ$ wale fluid mein hai (mass packed per cubic metre) aur stickiness $μ$ (viscosity — fluid kitni strongly sliding resist karta hai).

Figure — Similarity — geometric, kinematic, dynamic; Reynolds similarity

Step 2 — INERTIA force estimate karo ("keep going" push)

Yeh tool kyun — exactly solve karne ki jagah estimate kyun karein? Hum exact force nahi chahte, sirf yeh chahte hain ki yeh size aur speed ke saath kaise badhta hai. Do aisi estimates ka ratio hi similarity ko chahiye, aur estimates dead simple hain: har quantity ko "roughly is size ka" se replace karo.

HOW, term by term. Inertia force $=$ mass $\times$ acceleration.

$F_{i} \sim mass of cube ρ L^{3} \times acceleration \frac{V ^{2}}{L} = ρ V^{2} L^{2}$

$ρ L^{3}$ — cube ka volume $\approx L^{3}$ hai (side $L$ ka ek box), aur mass = density × volume.
$\frac{V ^{2}}{L}$ — acceleration hai change of speed ÷ time. Yahan sabse natural time yeh hai ki fluid ko body cross karne mein kitna waqt lagta hai: $t \sim L / V$ . Toh acceleration $\sim V \div (L / V) = V^{2} / L$ .
$\sim$ symbol ka matlab hai "ki tarah badhta hai / same size ka," exact constants jaise $\frac{1}{2}$ ko ignore karke.

PICTURE. Bada lavender arrow inertia push hai, cube ko aage drive karta hai.

Step 3 — VISCOUS force estimate karo ("stickiness drag")

Kyun. Sirf inertia half story hai: ek real fluid sliding resist karta hai. "Keep going" (inertia) aur "stickiness slows me" (viscous) ke beech ki competition hi decide karti hai ki flow ka poora character kya hoga — smooth ya turbulent, thin wake ya fat wake.

HOW, term by term. Viscous force $=$ shear stress $\times$ area.

$F_{v} \sim shear stress τ μ \frac{V}{L} \times contact area L^{2} = μ V L$

$\frac{V}{L}$ — velocity gradient: speed $V$ se $0$ tak $\sim L$ thickness ke across change hoti hai, toh "speed profile ki steepness" $V / L$ hai.
$μ \frac{V}{L}$ — viscosity times woh steepness deta hai shear stress $τ$ (rubbing ke unit area per force).
$L^{2}$ — us face ka area jahan rubbing hoti hai.

PICTURE. Fluid ki layers ek doosre ke past slide karti hain; coral arrows beech wali layer pe drag dikhate hain, steeper gradient → bada drag.

Step 4 — Dono forces divide karo → Reynolds number paida hota hai

Subtract kyun nahi, divide kyun? Similarity ko ratios se matlab hai, raw sizes se nahi. Ek ratio dimensionless hota hai — uske koi units nahi hote — isliye yeh "3.5" hi rehta hai chahe metres mein measure karo ya miles mein. Do flows jo same ratio rakhte hain woh same balance regime mein hain. Yahi poora trick hai.

HOW, term by term.

$Re = \frac{F _{i}}{F _{v}} = \frac{ρ V ^{2} L ^{2}}{μ V L} = \frac{ρ V L}{μ}$

Numerator $ρ V^{2} L^{2}$ — Step 2 se inertia.
Denominator $μ V L$ — Step 3 se viscous.
$V$ ki ek power aur $L$ ki ek power top-and-bottom cancel hoti hai, $ρ V L / μ$ bachta hai.

PICTURE. Steps 2–3 ke dono arrows, side by side, ek single dial mein jaate hain jis par $Re$ likha hai: bada Re = inertia jeeata hai (turbulent), chhota Re = stickiness jeeata hai (smooth).

Step 5 — Re match karne se POORA flow copy kyun hota hai

Yeh magic prove kyun karta hai. Agar do geometrically similar flows ka same $Re$ ho, toh unke stretched-unit equations letter-for-letter identical hain. Same equation + same shape of boundary ⇒ same dimensionless solution. Aur kuch differ nahi kar sakta.

HOW. Un chaar rulers ke saath, non-dimensional momentum equation kuch aisa likhta hai:

$\frac{\partial u ~}{\partial t ~} + (\tilde{u} \cdot \tilde{\nabla}) \tilde{u} = - \tilde{\nabla} \tilde{p} + \frac{1}{Re} \tilde{\nabla}^{2} \tilde{u}$

$\frac{\partial u ~}{\partial t ~}$ aur $(\tilde{u} \cdot \tilde{\nabla}) \tilde{u}$ — inertia terms (speed $V$ ki units mein, time $L / V$ ki units mein). Koi leftover constant nahi, exactly isliye kyunki humne woh do rulers choose kiye.
$- \tilde{\nabla} \tilde{p}$ — pressure push, constant-free kyunki humne pressure yardstick $ρ V^{2}$ purpose se choose kiya taaki iska coefficient $1$ ban jaaye.
$\frac{1}{Re} \tilde{\nabla}^{2} \tilde{u}$ — viscous term, aur iska sirf ek dial hai $1/ Re$ .

Re match karo → dono equations coincide ho jaate hain → same streamlines, same pressure pattern, same drag coefficient (Drag Coefficient dekho).

PICTURE. Do boxes alag physical size ke ek identical dimensionless picture pe collapse ho jaate hain jab Re match ho jaata hai — streamlines perfectly overlap karte hain.

Step 6 — Scaling rule padhna (same fluid)

Kyun. Yeh woh actual cheez hai jo ek engineer compute karta hai: wind tunnel kitni tezi se chalani hai.

HOW. Same fluid ke saath, $ρ$ aur $μ$ dono sides pe identical hain aur cancel ho jaate hain:

$\frac{ρ V _{m} L _{m}}{μ} = \frac{ρ V _{p} L _{p}}{μ} ⟹ V_{m} L_{m} = V_{p} L_{p} ⟹ V_{m} = V_{p} \frac{L _{p}}{L _{m}}$

$V_{m} L_{m} = V_{p} L_{p}$ — speed aur size ka product preserve hona chahiye.
Toh ek chhota model ( $L_{m} < L_{p}$ ) ko zyada speed chahiye. $1/5$ model → $5 \times$ speed.

PICTURE. Ek chhota model aur ek bada prototype: chhote ka speed arrow paanch guna lamba hai, $V L$ constant rakhte hue.

Step 7 — Degenerate aur edge cases

Case A — alag fluids. Tab $ρ, μ$ cancel nahi hote. Teeno factors rakhho: $V_{m} = V_{p} \frac{L _{p}}{L _{m}} \frac{ν _{m}}{ν _{p}}$

Case B — free surface hai (waves). Ab gravity matter karta hai, isliye ruling number Froude number ban jaata hai, Reynolds nahi.

Re-matching aur Froude-matching ek doosre se ladte hain (Re chahta hai $V \propto 1/ L$ , Fr chahta hai $V \propto L$ ). Ek fluid ke saath dono satisfy nahi kar sakte, isliye dominant force choose karni padti hai.

Case C — $μ \to 0$ (inviscid limit). Tab $Re \to \infty$ aur Step 5 ka $1/ Re$ term disappear ho jaata hai. Viscosity sirf ek razor-thin Boundary Layer mein survive karti hai jo surface se chipki hoti hai — baaki flow ise ignore karta hai.

Case D — bahut slow / tiny flow, $Re \to 0$ . Stickiness dominate karti hai; inertia negligible hai ("creeping flow"). Streamlines perfectly reversible ho jaate hain.

PICTURE. $lo g Re$ axis ke saath ek regime strip: creeping → laminar → transition → turbulent, charo cases mark kiye hue.

Ek-picture summary

Sab kuch ek canvas pe: cube pe do forces → unka ratio Re → stretched-unit equation sirf $1/ Re$ ke saath → matched Re flow copy karta hai → scaling rule $V_{m} = V_{p} (L_{p} / L_{m}) (ν_{m} / ν_{p})$ .

Recall Feynman retelling — poori walkthrough ek 12-saal ke bachche ko explain karo

Paani ke ek tiny cube ki picture banao. Do cheezein uske liye ladhti hain: momentum ("main zoom karta rehna chahta hoon!") aur stickiness ("mere aas-paas ka paani mujhe drag karta hai"). Humne guess kiya ki har push kitna bada hai sirf size aur speed use karke. Jab hum zoom-push ko drag-push se divide karte hain, saare units pighal jaate hain aur hum ek plain number ke saath bach jaate hain — Reynolds number. Yeh flow ka "flavour" jaisa hai. Ab yeh miracle hai: agar tum paani ka rulebook body ki apni size aur speed ko ruler ki tarah use karke rewrite karo, toh poora rulebook us ek number ke siwa sab kuch bhool jaata hai. Toh ek toy model aur ek giant real cheez, agar woh same Reynolds number aur same shape share karein, identical rulebook follow karte hain — unke swirls perfectly line up ho jaate hain. Chhote model ko badi cheez ka number share karwane ke liye, tum chhote pe tezi se hawa chalate ho (paanch guna chhota ⇒ paanch guna tez). Aur agar fluids alag hain, toh tum yeh bhi multiply karte ho ki har fluid kitna sticky hai. Edges pe dhyan rakho: waves ko ek alag number chahiye (Froude), super-fast flow stickiness ignore karta hai siwa ek paper-thin skin ke, aur super-slow flow poori stickiness hai.