Last updated: 2026-06-15

Reynolds Number

The Reynolds number is the single most useful number in fluid mechanics: a dimensionless ratio that tells you whether a flow will be smooth and orderly or chaotic and turbulent. It is why a thin trickle of water runs clear while a fast river churns, why blood flows quietly through capillaries but noisily through a narrowed artery, and why engineers can test a tiny model aircraft in a wind tunnel and trust the result. This calculator finds it from the fluid's density and viscosity, the flow speed, and a characteristic length.

Where the formula comes from

A flow is a contest between two forces. Inertia carries the fluid forward and amplifies any disturbance; viscosity is internal friction that smooths disturbances out. The Reynolds number is simply their ratio, $Re = \rho v L / \mu$. When viscosity dominates ($Re$ small) the flow stays laminar; when inertia dominates ($Re$ large) small disturbances grow into turbulence.

Formula

It can also be written $Re = vL/\nu$ with the kinematic viscosity $\nu = \mu/\rho$.

Worked example

Water flowing at $v = 1\ \text{m/s}$ through a pipe of diameter $L = 0.05\ \text{m}$, with $\rho = 1000\ \text{kg/m}^3$ and $\mu = 0.001\ \text{Pa·s}$:

That is well above 4000, so the flow is firmly turbulent — as you would expect for tap water moving briskly through a pipe.

Reading the result

For flow in a round pipe the conventional bands are:

These thresholds apply to pipe flow. Flow over a flat plate, around a sphere or through an open channel each transitions at its own critical value, so always note which geometry — and which characteristic length — you are using.

Limitations

The Reynolds number predicts the type of flow, not its detailed pattern, and the transition values are guidelines rather than hard limits — a very smooth, carefully fed pipe can stay laminar to much higher $Re$. The formula assumes a single Newtonian fluid with constant properties; it does not cover compressible or multiphase flows, and the right characteristic length depends entirely on the geometry.