The Reynolds number, explained

What it measures

Inertia keeps a fluid parcel moving the way it was going. Viscosity smooths out differences between neighbouring parcels. When viscosity wins, at low Re, the flow is laminar: orderly layers that slide past each other. When inertia wins, small disturbances grow, the flow becomes turbulent, and mixing, drag and heat transfer all go up.

μ is the dynamic viscosity in Pa·s and ν = μ/ρ is the kinematic viscosity in m²/s. Use whichever one your data table gives, with the matching form of the equation.

The characteristic length

L is not universal. In a pipe it is the diameter. On a flat plate it is the distance from the leading edge. On a wing it is the chord, and around a cylinder it is the diameter. The thresholds below only mean anything with the length they were defined for.

In a pipe, flow is reliably laminar below about 2,300 and fully turbulent above about 4,000, with a transition band between. On a flat plate the boundary layer usually turns turbulent around 5 × 10⁵. These are typical values, not laws: a very smooth, quiet inlet can stay laminar far longer.

Why it is everywhere

Two geometrically similar flows with the same Reynolds number behave the same way. That is dynamic similarity, and it is why a scale model in a wind tunnel tells you something about the full-size car. It is also why friction factors, drag coefficients and heat transfer correlations are all plotted against Re.

A worked number

Water at 20 °C has ν ≈ 1.0 × 10⁻⁶ m²/s. In a 20 mm pipe at 1 m/s, Re = 1 × 0.02 / 10⁻⁶ = 20,000. That is well into turbulence, which is normal for household plumbing. To keep it laminar you would need to slow it to about 0.1 m/s.

Common mistakes

• Using the wrong characteristic length, for example a pipe’s length instead of its diameter.
• Mixing dynamic and kinematic viscosity, which puts Re off by a factor of the density, a thousand for water.
• Treating 2,300 as a sharp switch. Transition depends on the inlet, the wall roughness and the disturbances.