Smooth or chaotic?
Open a tap a little and the stream is clear and glassy; open it fully and it turns rough and noisy. The same switch happens in pipes, rivers, air ducts and blood vessels, and it changes everything from pumping costs to heat transfer. The Reynolds number tells you which side of the switch a flow is on.
The formula
Re = ρ × v × L ÷ μ = v × L ÷ ν
- ρ is the density of the fluid (kg/m³).
- v is the flow speed (m/s).
- L is the characteristic length: the inner diameter of a pipe, the distance from the leading edge of a plate, or the diameter of a sphere.
- μ is the dynamic viscosity (Pa·s), and ν = μ ÷ ρ the kinematic viscosity.
Because the units cancel, Re has no dimension. Two flows with the same Reynolds number and the same shape behave alike, which is why engineers can test small models in wind tunnels.
Thresholds by geometry
| Geometry | Laminar | Turbulent |
|---|---|---|
| Inside a pipe | below about 2,300 | above about 4,000 |
| Along a flat plate | below about 500,000 | above about 500,000 |
| Around a sphere | creeping flow below 1 | turbulent wake above about 1,000 |
Example
Water at 68 °F (density 998 kg/m³, viscosity 1.0 mPa·s) flows at 3 ft/s (0.91 m/s) through a pipe with a 1-inch (25.4 mm) inner diameter:
Re = 998 × 0.914 × 0.0254 ÷ 0.001002 ≈ 23,100
The flow is well into the turbulent range. To keep it laminar in the same pipe, the water would have to flow slower than about 0.09 m/s (0.3 ft/s).
Friction in pipes
The calculator also gives the Darcy friction factor used in pressure-drop calculations. For laminar flow it is exactly 64 ÷ Re. For turbulent flow it uses the Haaland formula for a smooth pipe; real pipes are rougher, especially old steel, so their friction factor and pressure drop are higher. In the example, f ≈ 0.025.
Typical viscosities at room temperature
Water is about 1.0 mPa·s and air 0.018 mPa·s, but air is so light that its kinematic viscosity is about 15 times higher than water’s. Olive oil is about 80 mPa·s and glycerin about 1,400 mPa·s. Viscosity falls quickly as liquids warm up, so use values for the actual temperature when precision matters.