Reynolds Number: Predicting Flow Regime
The Reynolds number (Re) is the ratio of inertial to viscous forces. It predicts whether flow will be smooth and orderly (laminar) or chaotic and mixing (turbulent).
Formula
Re = ρvD / μ = vD / ν
ρ = fluid density (kg/m³)
v = velocity (m/s)
D = characteristic length (pipe diameter, m)
μ = dynamic viscosity (Pa·s)
ν = kinematic viscosity = μ/ρ (m²/s)Flow Regimes (pipe flow)
Re < 2300: Laminar — smooth, layered
2300 < Re < 4000: Transitional — unstable
Re > 4000: Turbulent — chaotic, mixingWorked Example
Water (20°C): ρ=998 kg/m³, μ=0.001 Pa·s
25mm pipe, v=1.5 m/s:
Re = 998 × 1.5 × 0.025 / 0.001 = 37,425
→ Turbulent (need Moody chart for friction factor)Fluid Properties at 20°C
- Water: ν = 1.0×10⁻⁶ m²/s, μ = 0.001 Pa·s
- Air: ν = 1.5×10⁻⁵ m²/s, μ = 1.81×10⁻⁵ Pa·s
- Engine oil (SAE30): ν ≈ 100×10⁻⁶ m²/s
Calculate Reynolds number: Free Reynolds Number Calculator
Reynolds Number and Flow Regimes
Re = ρVD/μ = VD/ν, where ρ = density (kg/m³), V = velocity (m/s), D = characteristic length (m), μ = dynamic viscosity (Pa·s), ν = kinematic viscosity (m²/s). For pipe flow: Re < 2,300 — laminar (smooth, layered flow, Hagen-Poiseuille applies). 2,300 < Re < 4,000 — transitional. Re > 4,000 — turbulent (chaotic mixing, higher friction losses, better heat transfer). Water at 20°C: ν = 1×10⁻⁶ m²/s. Air at 20°C: ν = 1.5×10⁻⁵ m²/s (15× more viscous kinematically than water).
Applications
- Pipe flow design: Confirming flow regime for friction factor selection (Moody chart).
- Aircraft aerodynamics: Scale model wind tunnel tests require matching full-scale Re. A 1:10 model at 10× velocity gives the same Re as the full aircraft.
- Mixing tanks: Impeller Re determines mixing regime — turbulent mixing (Re > 10,000) ensures uniform blending.
- Filtration: Laminar flow through filter media is governed by Darcy's Law; turbulent flow through coarse media requires different modelling.
Frequently Asked Questions
Why does Reynolds number determine flow behaviour?
Re = inertial forces / viscous forces. At low Re, viscosity dominates and damps disturbances — flow remains smooth and laminar. At high Re, inertia dominates and disturbances amplify into turbulent fluctuations. This is why honey (high viscosity, low Re) always flows laminarly and why large fast-moving pipes (low viscosity, high velocity) are always turbulent. The Reynolds number is dimensionless, allowing comparison across different fluids, velocities, and scales.
What is dynamic similarity and why is it used?
Two flows are dynamically similar if they have the same Reynolds number (and other relevant dimensionless groups). This allows scale model testing: a 1:20 scale aircraft model with air velocity 20× the full-scale speed has the same Re. Forces and pressures measured on the model scale predictably to the full-size aircraft. This is the principle behind wind tunnel and water channel testing in aeronautics, shipbuilding, and civil engineering.
Is there a single critical Reynolds number for transition?
No exact single value — transition depends on inlet conditions, surface roughness, freestream turbulence, and geometry. The 2,300/4,000 bounds for pipe flow are approximate. For flat plate boundary layers, transition occurs at Re_x ≈ 5×10⁵ (x = distance from leading edge). For flow around a sphere, drag crisis (sudden drop) occurs at Re ≈ 3×10⁵. Carefully controlled laminar flow in pipes can persist up to Re ≈ 20,000 in the lab.