is the ratio of inertial forces (v_sρ) to viscous forces (μ/L) and is used to determine whether a flow will be laminar or turbulent. It is the most important dimensionless number in fluid dynamics and provides a criterion for determining dynamic similitude. When two similar objects in perhaps different fluids with possibly different flowrates have similar fluid flow around them, they are said to be dynamically similar.

where:

• v_s - mean fluid velocity,
• L - characteristic length (equal to diameter (2r)
       if a cross-  section is circular),
• μ - (absolute) dynamic fluid viscosity,
• ν - kinematic fluid viscosity: ν = μ / ρ,
• ρ - fluid density

Laminar flow occurs at low Reynolds numbers (Re<2100), where viscous forces are dominant, and is characterized by smooth, constant fluid motion, while turbulent flow, on the other hand, occurs at high Reynolds numbers (Re>4000) and is dominated by inertial forces, producing random eddies, vortices and other flow fluctuations.

The transition between laminar and turbulent flow is often indicated by a critical Reynolds number (Re_crit), which depends on the exact flow configuration and must be determined experimentally. Within a certain range around this point there is a region of gradual transition where the flow is neither fully laminar nor fully turbulent, and predictions of fluid behaviour can be difficult. For example, within circular pipes the critical Reynolds number is generally accepted to be 2300, where the Reynolds number is based on the pipe diameter and the mean velocity v_s within the pipe, but engineers will avoid any pipe configuration that falls within the range of Reynolds numbers from about 2000 to 4000 to ensure that the flow is either laminar or turbulent.