Reynolds Number Formula
Calculate the Reynolds number using Re = ρvL/μ.
Determine whether fluid flow is laminar or turbulent in pipes and channels.
The Formula
The Reynolds number is a dimensionless quantity that predicts the flow regime of a fluid.
Low Reynolds numbers indicate smooth, laminar flow.
High Reynolds numbers indicate chaotic, turbulent flow.
Variables
| Symbol | Meaning |
|---|---|
| Re | Reynolds number (dimensionless) |
| ρ | Fluid density (kg/m³) |
| v | Flow velocity (m/s) |
| L | Characteristic length — typically pipe diameter (metres, m) |
| μ | Dynamic viscosity of the fluid (Pa·s or kg/(m·s)) |
Flow Regime
- Re < 2,300 — Laminar flow (smooth, predictable)
- 2,300 < Re < 4,000 — Transitional flow (unstable, may switch between laminar and turbulent)
- Re > 4,000 — Turbulent flow (chaotic, with eddies and mixing)
Example 1
Water (ρ = 1000 kg/m³, μ = 0.001 Pa·s) flows at 0.5 m/s through a pipe with diameter 0.05 m. Find Re.
Re = ρvL / μ
Re = (1000 × 0.5 × 0.05) / 0.001
Re = 25 / 0.001
Re = 25,000 — Turbulent flow
Example 2
Oil (ρ = 900 kg/m³, μ = 0.1 Pa·s) flows at 0.2 m/s through a 0.03 m diameter tube. Find Re.
Re = ρvL / μ
Re = (900 × 0.2 × 0.03) / 0.1
Re = 5.4 / 0.1
Re = 54 — Laminar flow
When to Use It
Use the Reynolds number when you need to:
- Determine whether flow in a pipe or channel is laminar or turbulent
- Select appropriate friction factor equations for pressure drop calculations
- Scale up laboratory experiments to real-world systems
- Design piping systems, heat exchangers, and fluid transport systems
The characteristic length L depends on the geometry.
For pipes, use the internal diameter. For flat plates, use the plate length.
Key Notes
- Formula: Re = ρvD/η = vD/ν: ρ is fluid density, v is velocity, D is the characteristic length (pipe diameter for internal flow, chord for airfoils), η is dynamic viscosity, and ν = η/ρ is kinematic viscosity. Re is dimensionless.
- Flow regimes in a pipe: Re < 2,300: laminar (smooth, layered flow, low friction losses). 2,300–4,000: transitional. Re > 4,000: turbulent (chaotic mixing, higher friction, better heat and mass transfer). The exact transition depends on pipe roughness and inlet conditions.
- Geometric similarity: Two flows with the same Re are dynamically similar — they behave identically when scaled. This is why scale models in wind tunnels can predict full-scale aircraft behavior, provided Re is matched.
- Turbulent flow friction is higher: In turbulent flow, the Darcy-Weisbach friction factor f depends on Re and wall roughness (Moody diagram). Higher Re typically means higher f and greater pressure drop for the same flow rate — a critical factor in pump sizing.
- Applications: Reynolds number governs pipe flow design (laminar vs turbulent friction calculations), aerodynamic body shaping, heat exchanger design (turbulence improves heat transfer), microfluidics (Re << 1 means purely laminar flow), and mixing reactor design.