Similarity — geometric, kinematic, dynamic; Reynolds similarity: Foundations

2.2.27 · D1 · Physics › Fluid Mechanics › Similarity — geometric, kinematic, dynamic; Reynolds similar

Parent note mein $L$ , $V$ , $ρ$ , $μ$ , $ν$ , "shear stress", "dimensionless", "force ratio", aur $Re$ aise use hote hain jaise tum pehle se jaante ho. Yahan hum inhe ek-ek karke earn karenge, aur ek aisa order rakhenge jahan har nayi cheez sirf pehle wali cheez par depend kare.

1. Length $L$ — flow ki "kitni badi hai"

Yeh picture: neeche di gayi figure mein ek full-size car aur uski half-size model dikhti hai. Ek red double-arrow $L_{p}$ (prototype ki length) mark karta hai aur ek violet double-arrow $L_{m}$ (model ki length) — tum jo bhi length apne "ruler" ki tarah chunte ho, baaki sabhi lengths usse compare ki jaati hain.

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

Figure s01 — Alt-text: a large magenta car labelled $L_{p}$ beside a violet half-size copy labelled $L_{m}$ ; both lengths carry double-headed arrows, and an orange label states $L_{r} = L_{m} / L_{p} = 1/2$ .

Topic ko yeh kyun chahiye. Similarity scaling ke baare mein hai — ek real object ko model tak shrink karna. Tum "1/5th size" nahi keh sakte jab tak tumne "size" measure karne ke liye ek length pick nahi ki. Woh chosen length $L$ hai, aur dono versions ka ratio scale factor hai.

2. Velocity $V$ — "kitna fast"

Yeh picture: moving fluid par draw kiya gaya ek arrow. Uski length = speed, uski direction = fluid kis tarah jaata hai. "Velocity" aisa hi dikhta hai.

Topic ko yeh kyun chahiye. Flow mein forces speed par depend karti hain (fast air slow air se zyada push karti hai). Flow ko scale karne ke liye, hume uski speed track karni hogi — isliye $V$ har formula mein aata hai. Velocities ka scale $V_{r} = V_{m} / V_{p}$ hai.

3. Force aur uski unit, Newton — kuch bhi weigh karne se pehle

Topic ko yeh kyun chahiye. Aage sab kuch — inertia force, viscous force, gravity force, shear stress $τ$ , drag $F_{D}$ — newtons ya pascals mein measure hota hai. Inhe abhi base units se define karna matlab hai ki koi bhi symbol ( $N$ , $Pa$ ) baad mein unexplained nahi aayega.

4. Density $ρ$ — "har box mein kitna bhaari"

Yeh picture: ek $1 m \times 1 m \times 1 m$ box fluid se bhara hua, scale par rakha hua. Scale par jo number aata hai woh $ρ$ hai.

Topic ko yeh kyun chahiye. Newton's law hai force = mass × acceleration. Fluid ke ek chunk ka mass pane ke liye hume jaanna hoga ki fluid per unit volume kitna bhaari hai — woh $ρ$ hai. Bhaari fluid ⇒ zyada inertia ⇒ push karna mushkil.

5. Gravitational acceleration $g$ — "cheezein kitni tezi se girti hain"

Yeh picture: ek seb chhoda gaya, uska downward-speed arrow har second same amount $g$ se lamba hota jaata hai.

Topic ko yeh kyun chahiye. Fluids ka weight hota hai, aur weight = mass × $g$ . Jab bhi kisi flow mein free surface ho (paani par lahrein, ship ka wake, spillway), gravity us surface ko wapas neeche kheenchti hai aur flow ko shape deti hai. Gravity force measure karne ke liye hume $g$ jaanna zaroori hai. Yeh neeche gravity-force estimate mein aur Froude number mein aata hai.

6. Viscosity $μ$ — "kitna sticky/gooey"

Ise precise banane ke liye hume ek aur picture chahiye: shear.

Figure s02 — Alt-text: stacked horizontal layers of fluid above a navy wall; magenta velocity arrows grow longer with height $y$ (zero at the wall, largest at the top), and an orange $y$ -axis and a caption mark the velocity gradient $d V / d y$ as how fast the arrows lengthen.

Topic ko yeh kyun chahiye. Reynolds similarity ka poora point inertia (fluid ki momentum se push) aur viscosity (fluid ki internal friction jo ise slow karti hai) ke beech contest hai. $μ$ aur $τ$ ke bina contest mein measure karne ke liye koi "friction" side hi nahi hai.

7. "Dimensionless" — kuch numbers mein units kyun nahi hote

Yeh picture: sochho ek ratio jaise "do guna lamba." Woh "2" same hai chahe tum feet mein measure karo ya metres mein — units division mein cancel ho gayi. Dimensionless numbers exactly wohi cancel-everything ratios hain.

Topic ko yeh kyun chahiye. Similarity ka magic sentence hai "dimensionless number match karo aur dono flows same hain." Yeh tabhi kaam karta hai kyunki dimensionless number unit-free hota hai — toh ek chota model aur ek giant prototype, bilkul alag scales par measure kiye gaye, honestly ek hi value share kar sakte hain.

8. Force ratios — dynamic similarity ka dil

Formula Inertia force

$F_{i} \sim ρ V^{2} L^{2}$ step by step banana Hum kya karte hain: inertia force = mass × acceleration (Newton's law), dimensions se size up kiya. ( $\sim$ ka matlab hai "scales like" — same size, exact factors jaise $\frac{1}{2}$ ignore kiye.)

Mass: size $L$ ke ek fluid chunk ka volume $\sim L^{3}$ hai, toh mass $\sim ρ L^{3}$ .
Acceleration — $V^{2} / L$ kyun? Acceleration hai speed mein change ÷ time liya. Fluid body ke paas se guzarta hai toh woh apni speed $V$ order ki amount se change karta hai us distance $L$ par jo woh body cross karne mein leta hai. Us distance ko speed $V$ par cross karne ka time $t \sim L / V$ hai. Toh $acceleration \sim \frac{change in speed}{time} \sim \frac{V}{L / V} = \frac{V ^{2}}{L} .$ Yahi ek timescale available hai — crossing time $L / V$ — isliye characteristic acceleration $V^{2} / L$ nikalta hai (sochho: ek particle distance $L$ par decelerate kar raha hai).
Multiply: $F_{i} \sim ρ L^{3} \cdot \frac{V ^{2}}{L} = ρ V^{2} L^{2}$ .

Neeche di gayi figure yeh sab ek literal tug-of-war ke roop mein draw karti hai: inertia ek taraf kheenchti hai, viscosity (aur, alag se, gravity) wapas kheenchti hai. Dono pulls ka ratio woh dimensionless number hai jis par hum dhyan dete hain.

Figure s03 — Alt-text: a rope tug-of-war; a magenta block labelled inertia $F_{i} \sim ρ V^{2} L^{2}$ pulls left, a violet block labelled viscous $F_{v} \sim μ V L$ pulls right; orange captions give $Re = F_{i} / F_{v} = ρ V L / μ$ and $Fr = F_{i} / F_{g} = V^{2} / (g L)$ , with notes that big Re means inertia wins and small Re means viscosity wins.

Topic ko dono kyun chahiye. Alag flows alag contests se rule hote hain. Submerged bodies aur pipes → inertia vs viscosity → $Re$ match karo. Free-surface flows → inertia vs gravity → $Fr$ match karo. Har ratio dimensionless hai, toh model aur prototype ke beech match ho sakta hai.

Formula Drag coefficient

$C_{D}$ — force ko dimensionless banana $C_{D} = \frac{F _{D}}{\frac{1}{2} ρ V ^{2} A}$ $A$ = ek reference area ( $\sim L^{2}$ ). $\frac{1}{2}$ kahan se aata hai? Quantity $\frac{1}{2} ρ V^{2}$ dynamic pressure hai — woh pressure jo ek moving fluid ruk jaane par produce karta. Yeh kinetic energy per unit volume se aata hai: $ρ$ density aur $V$ speed wale fluid ka ek chunk $\frac{1}{2} m V^{2}$ kinetic energy per unit volume $= \frac{1}{2} ρ V^{2}$ carry karta hai (wahi $\frac{1}{2}$ jaise $\frac{1}{2} m V^{2}$ mein). Us pressure ko area $A$ se multiply karne par ek natural "reference force" milti hai jisse real drag divide karte hain. $F_{D}$ (units $N$ ) ko $\frac{1}{2} ρ V^{2} A$ (bhi units $N$ ) se divide karne par pure number milta hai.

Jab do flows same $Re$ share karte hain, woh same $C_{D}$ share karte hain, aur yahi woh tarika hai jisse model measurements prototype drag predict karte hain.

9. Yeh sab topic ko kaise feed karta hai

Jab yeh foundations in place hain, parent par jao: Similarity — geometric, kinematic, dynamic; Reynolds similarity (index 2.2.27). Related deeper stops: Reynolds Number, Drag Coefficient, Froude Number, Buckingham Pi Theorem, Navier-Stokes Equations, Boundary Layer.

Equipment checklist

Right side cover karo aur khud ko test karo.

Subscript

m

p

ka kya matlab hai?

m

= model (chota test piece),

p

= prototype (asli full-size cheez).

Scale factor

L_{r}

kya hai?

Model length aur prototype length ka ratio,

L_{r} = L_{m} / L_{p}

Base SI units mein ek newton kya hota hai?

1 N = 1 kg \cdot m / s^{2}

— woh push jo

1

kg ko har second

1

m/s accelerate kare.

Ek pascal kya hai?

1 Pa = 1 N / m^{2} = 1 kg / (m \cdot s^{2})

— force spread over area.

Density

ρ

ki units kya hain?

Kilograms per cubic metre,

kg / m^{3}

g

kya hai aur uski value/units kya hain?

Gravitational acceleration,

\approx 9.81 m / s^{2}

— girne ki speed har second kitni badhti hai.

Seedhe shabdon mein viscosity

μ

kya hai?

Fluid layers ke beech internal "grip" — ek layer doosri ko kitni strongly drag karti hai.

Velocity gradient

d V / d y

kya measure karta hai?

Flow speed kitni tezi se change hoti hai jab tum crosswise, wall se door jaate ho.

Newton's law of viscosity likho.

τ = μ \frac{d V}{d y}

(shear stress = viscosity × velocity gradient).

Characteristic acceleration

V^{2} / L

kyun scale karti hai?

Speed

\sim V

amount se crossing time

t \sim L / V

par change hoti hai, toh

a \sim V / (L / V) = V^{2} / L

Derive karo ki viscous force

μ V L

kyun scale karti hai.

Force = stress × area

\sim (μ V / L) \times L^{2} = μ V L

Kinematic viscosity

ν

μ

se kya rishta hai?

ν = μ / ρ

, units

m^{2} / s

"Dimensionless" ka kya matlab hai?

Ek pure number jismein saari units cancel ho gayi hain, kisi bhi measuring system mein same.

Hum raw forces nahi, force RATIOS kyun compare karte hain?

Model aur prototype ke beech raw forces bahut alag hoti hain, lekin unke ratios equal banaye ja sakte hain.

Re

ke chaar equivalent ways likho.

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

Froude number likho aur kab use karte hain.

Fr = \frac{F _{i}}{F _{g}} = \frac{V ^{2}}{g L}

; use karo free-surface flows ke liye jahan gravity inertia se compete kare.

Drag force

F_{D}

kya hai?

Flow ke along-direction mein fluid body par jo push exert karta hai (pressure + shear dono), newtons mein.

C_{D}

mein

\frac{1}{2}

kahan se aata hai?

Dynamic pressure

\frac{1}{2} ρ V^{2}

se, yaani kinetic energy per unit volume (

\frac{1}{2} m V^{2}

per volume).

Physically bada

Re

kya batata hai?

Inertia viscosity par dominate karti hai — flow turbulence ki taraf jaati hai.

Drag coefficient

C_{D}

define karo.

C_{D} = \frac{F _{D}}{\frac{1}{2} ρ V ^{2} A}

, drag ka ek dimensionless measure.