Similarity — geometric, kinematic, dynamic; Reynolds similarity

2.2.27 · Physics › Fluid Mechanics

2. Dynamic similarity ko first principles se derive karna

HOW ratios banate hain — dimensions se har force estimate karo.

Ek characteristic length $L$ , velocity $V$ , density $ρ$ , viscosity $μ$ lo.

Inertia force = mass × acceleration. Mass $\sim ρ L^{3}$ , acceleration $\sim \frac{V}{t} \sim \frac{V}{L / V} = \frac{V ^{2}}{L}$ . $F_{i} \sim ρ L^{3} \cdot \frac{V ^{2}}{L} = ρ V^{2} L^{2}$ Yeh step kyun? Humne time ko $L / V$ se replace kiya (body cross karne ka time), jo ek maatra available timescale hai.
Viscous force = shear stress × area. Stress $τ = μ \frac{d V}{d y} \sim μ \frac{V}{L}$ , area $\sim L^{2}$ . $F_{v} \sim μ \frac{V}{L} \cdot L^{2} = μ V L$

3. Reynolds similarity

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

Worked Example A — wind-tunnel speed

Ek car ( $L_{p} = 4$ m) $V_{p} = 30$ m/s pe chalti hai. Ek $1/5$ model ( $L_{m} = 0.8$ m) ko air mein test kiya jaata hai (same $ρ, μ$ ). $V_{m}$ nikalo.

Yeh step kyun? Koi free surface nahi, viscous drag matter karta hai → Reynolds similarity. $V_{m} = V_{p} \frac{L _{p}}{L _{m}} = 30 \times \frac{4}{0.8} = 150 m/s$
Kyun? Chota body → $ρ V L / μ$ same rakhne ke liye zyada tez chalaana padega. (Yeh high speed exactly isliye hai ki engineers aksar water tunnels ya pressurised tunnels use karte hain.)

Worked Example B — force scale karna

Re match hone par, drag coefficient $C_{D} = \frac{F _{D}}{\frac{1}{2} ρ V ^{2} A}$ dono ke liye equal hota hai. Maan lo model drag $F_{D, m} = 12$ N hai. Prototype drag nikalo. $\frac{F _{D, p}}{ρ _{p} V _{p}^{2} A _{p}} = \frac{F _{D, m}}{ρ _{m} V _{m}^{2} A _{m}}$ Same fluid aur $V_{m} = V_{p} / L_{r}$ , $A = L^{2}$ ke saath: $F_{D, p} = F_{D, m} \frac{V _{p}^{2} A _{p}}{V _{m}^{2} A _{m}} = F_{D, m} \cdot L_{r}^{2} \cdot \frac{1}{L _{r}^{2}} = F_{D, m} = 12 N$

Yeh surprising result kyun? Same fluid mein Re match hone par, model ki zyada speed exactly chote area ko compensate karti hai — forces equal ho jaate hain. (Agar fluids alag hoon, toh actual $ρ, V, A$ plug karo — equality assume mat karo.)

Worked Example C — different fluid

Ek pipe ( $D_{p} = 0.5$ m) mein water ( $ν_{p} = 1 \times 1 0^{- 6}$ m²/s) $V_{p} = 2$ m/s pe flow karti hai. Ek model air ( $ν_{m} = 1.5 \times 1 0^{- 5}$ m²/s) use karta hai, $D_{m} = 0.1$ m. $V_{m}$ nikalo. $\frac{V _{m} D _{m}}{ν _{m}} = \frac{V _{p} D _{p}}{ν _{p}} \Rightarrow V_{m} = V_{p} \frac{D _{p}}{D _{m}} \frac{ν _{m}}{ν _{p}} = 2 \cdot \frac{0.5}{0.1} \cdot \frac{1.5 \times 1 0 ^{- 5}}{1 \times 1 0 ^{- 6}} = 150 m/s$

$ν$ ratio kyun include kiya? General Re-matching mein teeno factors rehte hain; sirf jab fluids identical hoon tabhi $ν$ cancel hota hai.

4. Common mistakes (steel-manned)

5. Active recall

Recall Self-test (dekhne se pehle try karo)

Kinematic similarity ke liye geometric similarity kyun zaroori hai?
Inertia/viscous force ratio se Re derive karo.
Same fluid mein Re-matching ke under $V_{m}$ ka $V_{p}$ se kya relation hai?
Reynolds ki jagah Froude kab use karte hain?
Re match karna $C_{D}$ equal kyun bana deta hai?

Recall Feynman: ek 12-saal ke bachhe ko samjhao

Socho tum ek bade ship ke baare mein seekhne ke liye ek chota toy boat banate ho. Toy tabhi real ship jaisi behave karti hai jab woh same shape ho (geometric), paani same pattern mein ghume (kinematic), aur push-aur-drag forces same tarike se balance hoon (dynamic). Scientists ne ek single "magic number" dhundha, Reynolds number, jo speed, size, aur liquid kitni thick hai — yeh sab mix karta hai. Agar toy ka magic number ship ke magic number ke barabar ho, toh toy bilkul real ship ki tarah move karti hai — toh jo bhi tum toy pe measure karo, real ship ke liye scale up kar sako. Cool trick: choti toy ko apna magic number same rakhne ke liye real ship se zyada tez chalana padta hai!

Flashcards

Geometric similarity ka matlab hai

same shape — saari lengths ek constant factor

L_{r}

se scale hoti hain, angles preserve rehte hain.

Kinematic similarity ka matlab hai

same motion pattern — velocity/acceleration fields sirf ek constant scale se alag hote hain; geometric similarity zaroori hai.

Dynamic similarity ka matlab hai

model aur prototype mein corresponding force ratios equal hote hain; kinematic similarity imply karta hai.

Reynolds number ka physical meaning

inertia force aur viscous force ka ratio,

Re = ρ V L / μ = V L / ν

Inertia force scaling estimate

F_{i} \sim ρ V^{2} L^{2}

Viscous force scaling estimate

F_{v} \sim μ V L

Reynolds similarity criterion

viscous-dominated flows (koi free surface nahi) ke liye

Re_{m} = Re_{p}

match karo.

Same-fluid Re-matching se model speed milti hai

V_{m} = V_{p} (L_{p} / L_{m}) = V_{p} / L_{r}

(chota model → zyada tez).

Froude Reynolds ki jagah kab use karte hain

free-surface/gravity-wave flows (ships, spillways) ke liye;

Fr = V / g L

Re match karna

C_{D}

equal kyun banata hai

non-dimensional Navier–Stokes mein sirf

1/ Re

coefficient hai, toh equal Re → identical dimensionless solution.

Same fluid mein Re aur Fr saath match kyun nahi ho sakte

Re ko

V \propto 1/ L

chahiye, Fr ko

V \propto L

chahiye — yeh contradictory hai.

Connections

Reynolds Number — laminar/turbulent transition isi number ka use karta hai.
Buckingham Pi Theorem — dimensionless groups ka formal source.
Navier-Stokes Equations — non-dimensionalisation Re-similarity ko justify karta hai.
Drag Coefficient — dynamic similarity ke under preserve hone wali quantity.
Froude Number — free-surface flows ke liye complementary criterion.
Boundary Layer — Re uski thickness aur separation control karta hai.