the Reynolds number
/ RENN-uldz /
Why does syrup pour in a smooth ribbon while a river churns into eddies? Why does a tiny swimming bacterium feel water as thick as treacle, while a whale feels it as we do? The answer is a single dimensionless number that compares two competing effects in a flow: inertia, the fluid's tendency to keep barrelling along, versus viscosity, its internal stickiness that smooths motion out. That number is the Reynolds number.
It is defined as Re = (rho U L) / mu, equivalently U L / nu, where rho is density, U a typical flow speed, L a typical length scale (pipe diameter, wing chord, swimmer size), mu the dynamic viscosity, and nu = mu / rho the kinematic viscosity. The point is what Re measures in the Navier-Stokes equations: it is essentially the ratio of the size of the nonlinear inertial term (u dot grad) u to the size of the viscous term mu Laplacian u. Small Re (well below about 1) means viscosity wins: flow is laminar, smooth, and reversible-looking. Large Re (thousands and up) means inertia wins: flow becomes unstable, then turbulent. A pipe typically transitions to turbulence around Re of a few thousand.
The Reynolds number is the master key of fluid similarity: two flows with the same Re and the same shape behave the same way, no matter their absolute size or speed. That is exactly why a scale model of a plane in a wind tunnel predicts the real plane — you match Re. It also explains the strange physics of the very small: a bacterium lives at Re around 0.0001, where coasting is impossible and stopping is instantaneous, a world with no momentum to speak of.
Water (nu about 1e-6 m^2/s) flowing at U = 1 m/s through a pipe of diameter L = 0.05 m gives Re = U L / nu = (1)(0.05)/(1e-6) = 50000 — firmly turbulent. The same pipe with thick oil (nu a thousand times larger) gives Re = 50 — smooth, laminar flow.
Re = inertia / viscosity; the same geometry can be laminar or turbulent depending on Re.
There is no single universal Re at which flow turns turbulent; the transition value depends on geometry, surface roughness, and disturbances, so the famous 'around 2000-4000 in a pipe' is a rough range, not a law.