Reynolds Number
Calculate the Reynolds number Re = ρvD/μ to predict laminar or turbulent flow.
Critical threshold ~2300 for pipe flow.
The Formula
The Reynolds number is a dimensionless quantity that predicts whether fluid flow will be smooth and orderly (laminar) or chaotic and mixed (turbulent). It was introduced in 1883 by Osborne Reynolds, an Irish-born engineer working at the University of Manchester in the United Kingdom. His famous dye experiment showed that a thin stream of colored dye in a pipe remained straight and steady at low flow rates but broke apart into swirling eddies at higher flow rates.
The Reynolds number represents the ratio of inertial forces to viscous forces in a fluid. When viscous forces dominate (low Re), the fluid flows in smooth, parallel layers — this is laminar flow. When inertial forces dominate (high Re), the flow becomes chaotic and turbulent. For flow inside a circular pipe, the critical Reynolds number is approximately 2,300. Below this value, flow is laminar. Above about 4,000, flow is fully turbulent. Between 2,300 and 4,000 is a transition zone where either regime can occur.
The Reynolds number is one of the most widely used dimensionless numbers in engineering. It helps engineers design efficient piping systems, predict heat transfer rates, and scale laboratory models to full-size structures. A small-scale wind tunnel model and a full-size aircraft will behave similarly if they have the same Reynolds number, even though their physical sizes are very different.
The formula can also be written as Re = vD/ν, where ν (nu) is the kinematic viscosity (ν = μ/ρ). This form is sometimes more convenient because kinematic viscosity values are commonly tabulated for many fluids.
Variables
| Symbol | Meaning |
|---|---|
| Re | Reynolds number (dimensionless — no units) |
| ρ | Fluid density (kg/m³) — water ≈ 1,000, air ≈ 1.225 |
| v | Flow velocity (m/s) |
| D | Characteristic length (m) — pipe diameter for internal flow, object length for external flow |
| μ | Dynamic viscosity (Pa·s or kg/(m·s)) — water ≈ 0.001, air ≈ 0.0000181 |
Example 1
Water at 20°C flows through a 2.5 cm diameter pipe at 0.5 m/s. Is the flow laminar or turbulent? (ρ = 998 kg/m³, μ = 0.001 Pa·s)
Convert diameter: D = 2.5 cm = 0.025 m
Re = ρvD / μ = 998 × 0.5 × 0.025 / 0.001
Re = 12.475 / 0.001 = 12,475
Re ≈ 12,475 — well above 4,000, so the flow is fully turbulent
Example 2
Honey (ρ = 1,400 kg/m³, μ = 5 Pa·s) flows through a 3 cm pipe at 0.1 m/s. Is the flow laminar or turbulent?
Convert diameter: D = 0.03 m
Re = 1,400 × 0.1 × 0.03 / 5
Re = 4.2 / 5 = 0.84
Re ≈ 0.84 — extremely low, the flow is very much laminar (thick fluids resist turbulence)
When to Use It
Use the Reynolds number whenever you need to determine the flow regime or scale a fluid dynamics problem.
- Designing piping systems — laminar and turbulent flows have very different friction losses
- Sizing pumps and selecting pipe diameters for desired flow conditions
- Wind tunnel testing — matching Reynolds numbers ensures accurate scale models
- Predicting heat transfer rates (convection coefficients depend on flow regime)
- Calculating drag on vehicles, aircraft, and underwater structures