## The Reynolds Number

The Reynolds number Re is a dimensionless number, the ratio of inertial forces to viscous forces i.e. where

• is the mean relative velocity between object and fluid (m/s).

• is a characteristic linear dimension (m).

• is the dynamic viscosity of the fluid (kg/ms).

• is the density of the fluid (kg/m³).

Expressions such as those above are used in dimensional analysis, and the Reynolds number is used to distinguish between smooth/laminar or turbulent flow. Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion, while turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tends to produce chaotic eddies and vortices.

Reynolds numbers are not uniquely defined, but may be specific to a situation. They generally take into account density, viscosity, velocity and a length. For pipes the length may be the radius and for aircraft or ships, the length or width may be used. For flow in a pipe or a sphere moving in a fluid the internal diameter is generally used today. For fluids of variable density (e.g. compressible gases) or variable viscosity (non-Newtonian fluids) special rules apply.

For flow in a circular pipe or tube or flow around a sphere in a fluid (also falling bodies), the Reynolds number is generally defined as where:

• D is the diameter of the pipe (m).

• A is the pipe cross-sectional area (m²).

• V is the velocity of fluid relative to pipe or fluid (m/s) 