Playground
Streamlines in a pipe transition from laminar to turbulent as Reynolds number crosses ~2300. Adjust velocity and viscosity.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Reynolds Numberoutput Dimensionless flow regime indicator | dimensionless | 1 | 0 – 1000000 | |
| Density Fluid density | kg/m³ | M·L⁻³ | 1 – 13600 | |
| Velocity Characteristic flow velocity | m/s | L·T⁻¹ | 0 – 50 | |
| Characteristic Length Characteristic length scale (e.g., pipe diameter) | m | L | 0.0001 – 10 | |
| Dynamic Viscosity Fluid dynamic viscosity | Pa·s | M·L⁻¹·T⁻¹ | 0.00001 – 10 |
Deep dive
Derivation
Re comes from non-dimensionalizing the Navier-Stokes equation. The inertial term ρv·∇v scales as ρv²/L, and the viscous term μ∇²v scales as μv/L². Their ratio is (ρv²/L)/(μv/L²) = ρvL/μ ≡ Re. When Re ≫ 1, inertia dominates; when Re ≪ 1, viscosity dominates.
Experimental verification
Reynolds's original 1883 dye-injection experiment in glass pipes showed sharp laminar-to-turbulent transition near Re ≈ 2000. Repeated countless times since with hot-wire anemometry, PIV, and DNS simulations.
Common misconceptions
- Re = 2300 is NOT a universal turbulence threshold — it's specific to pipe flow
- Re depends on the chosen characteristic length L — different choices give different numbers for the same flow
- High Re does NOT mean 'fast' — it means 'inertially dominated', which can occur at any speed if μ is small enough
Real-world applications
- Pipe and duct design in plumbing and HVAC
- Aircraft and ship hydrodynamics scaling (wind tunnel models)
- Microfluidic device design (low-Re regime)
- Blood flow analysis in arteries
- Predicting drag coefficients on spheres and cylinders
Worked examples
Water in a 5 cm pipe at 1 m/s
Given:
- ρ:
- 1000
- v:
- 1
- L:
- 0.05
- μ:
- 0.001
Find: Re
Solution
Re = (1000 × 1 × 0.05) / 0.001 = 50000 → fully turbulent
Bacterium swimming in water
Given:
- ρ:
- 1000
- v:
- 0.00003
- L:
- 0.000002
- μ:
- 0.001
Find: Re
Solution
Re = (1000)(3e-5)(2e-6) / 0.001 = 6e-5 → deeply Stokes regime
Scenarios
What if…
- scenario:
- What if you doubled the pipe diameter at fixed velocity?
- answer:
- Re doubles. Larger pipes are more prone to turbulence at the same speed — which is why big rivers turbulate easily while capillaries stay laminar.
- scenario:
- What if you replaced water with honey (1000× more viscous)?
- answer:
- Re drops by 1000×. Honey at 1 m/s in a 5 cm pipe has Re ≈ 50 — well below the turbulence threshold, so it flows in smooth layers.
- scenario:
- What if you scale a model airplane down 100×?
- answer:
- To preserve Re, you must increase v×L by 100. Wind-tunnel testing uses high-speed flow or pressurized gas to match the full-scale Re — otherwise the flow physics differs.
Limiting cases
- condition:
- Re < 2300 (pipe)
- result:
- Laminar flow
- explanation:
- Smooth, layered streamlines — viscous forces dominate.
- condition:
- 2300 < Re < 4000
- result:
- Transitional
- explanation:
- Intermittent turbulent bursts — unpredictable regime.
- condition:
- Re > 4000
- result:
- Turbulent flow
- explanation:
- Chaotic eddies, enhanced mixing — inertial forces dominate.
- condition:
- Re → 0
- result:
- Stokes (creeping) flow
- explanation:
- Viscosity completely dominates — used for microfluidics and bacteria swimming.
Context
Osborne Reynolds · 1883
Reynolds injected dye into pipe flows and observed the transition from laminar streaks to turbulent swirls, identifying the dimensionless number that governs the transition.
Hook
When does smooth flowing water become a chaotic turbulent mess?
Water (ρ=1000 kg/m³, μ=0.001 Pa·s) flows through a 5 cm pipe at 1 m/s. Find Re and determine if the flow is laminar or turbulent.
Dimensions: [ρvL/μ] = (M·L⁻³)(L·T⁻¹)(L) / (M·L⁻¹·T⁻¹) = (M·L⁻¹·T⁻¹)/(M·L⁻¹·T⁻¹) = 1 ✓ (dimensionless)
Validity: Newtonian fluids; transition Re depends strongly on geometry (≈2300 for pipes, ≈5×10⁵ for flat plates).