04 // Interactive Formula Compendium // ENGINEERINGRe = (ρ · v · D) / μ = (v · D) / ν
Reynolds Number Formula (Fluid Regime)
The Reynolds number Re = ρvD/μ quantifies the ratio of fluid inertial momentum to dynamic viscosity; internal pipe flow is laminar for Re < 2,300 and fully turbulent above Re > 4,000.
Formula Reference
Main Formula
Re = (ρ · v · D) / μ = (v · D) / ν
Rearranged Forms
ν = μ / ρ
v_crit = (2300 · μ) / (ρ · D)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| Re | Reynolds Number | dimensionless |
| rho | Fluid Density | kg/m³ |
| v | Mean Flow Velocity | m/s |
| D | Hydraulic Diameter | m |
| mu | Dynamic Viscosity | Pa·s |
Inputs & Parameters
[kg/m³]
[m/s]
[m]
[Pa·s]
Calculated Result (Reynolds Number)
Re =74,850dimensionless
Formula Notes & Background
- Obtained by non-dimensionalizing the incompressible Navier-Stokes momentum equation using characteristic length D and velocity v.
- For non-circular ducts, substitute hydraulic diameter D_h = 4A / P_wetted.
Step-by-Step Examples
#1: Water at 20 °C (ρ = 998 kg/m³, μ = 1.00 × 10⁻³ Pa·s) flowing at 1.5 m/s inside a 50 mm pipe.
ρ = 998 kg/m³v = 1.5 m/sD = 0.05 mμ = 0.001 Pa·s
Re = (998 × 1.5 × 0.05) / 0.001 = 74.85 / 0.001
→ Re = 74,850 (Fully turbulent pipe flow)
#2: SAE 30 oil (ρ = 890 kg/m³, μ = 0.29 Pa·s) pumped at 0.8 m/s in a 25 mm hydraulic line.
ρ = 890 kg/m³v = 0.8 m/sD = 0.025 mμ = 0.29 Pa·s
Re = (890 × 0.8 × 0.025) / 0.29 = 17.8 / 0.29
→ Re ≈ 61.4 (Deeply laminar flow)
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Frequently Asked Questions
What are the Reynolds number thresholds in circular pipes?+
Laminar below Re ≈ 2,300, transitional between 2,300 and 4,000, and turbulent above Re ≈ 4,000.
What is kinematic viscosity ν?+
Kinematic viscosity ν = μ/ρ (measured in m²/s or Stokes) combines dynamic viscosity and density into a momentum diffusivity coefficient.
How is hydraulic diameter calculated for a rectangular duct?+
D_h = 4ab / (2(a+b)) = 2ab / (a+b), where a and b are the interior duct width and height.