# The Reynolds Number

The Reynolds number is not only used to characterize the flow patterns, such as laminar or turbulent flow, but also to determine the dynamic similitude between two different flow cases.

The latter is an important concept to ensure valid experimental or computational results of numerous scaled models for industrial applications.

The Reynolds number is the ratio of the inertial force to the viscous force, defined as (1)
$\mathrm{Re}=\frac{\rho \text{\hspace{0.17em}}u\text{\hspace{0.17em}}L}{\mu }$

where $\rho$ is the fluid density (kg/m3), $u$ the mean flow velocity (m/s), $L$ the characteristic length (m), for example, hydraulic diameter for internal flows, and $\mu$ the dynamic viscosity of the fluid (kg/(m-s).

The characteristic length in the Reynolds number can be anything convenient, as long as it is consistent, especially when comparing different geometries. For example, the radius and the diameter are both valid for spheres or circles in flows with the diameter primarily used by convention. When computing the Reynolds number for airfoils and wings, the chord length is often chosen over the span because the former is a representative function of the lift. For pipes, the diameter is the characteristic length. For rectangular pipes a suitable choice is the hydraulic diameter defined as (2)
${D}_{h}=\frac{4A}{P}$

where A is the cross-section area (m2) and P is the wetted perimeter (m). The wetted perimeter is the total perimeter of subject walls in contact with the flow.

Fluid flows are laminar when their Reynolds number is below a certain critical value and they are turbulent when they are larger than this critical value, termed the critical Reynolds number (Recr). This critical Reynolds number is between 2,300 and 4,000 for pipe flows.

The critical Reynolds number is also referred to as the transition Reynolds number. It varies widely depending on the conditions of surface roughness, flow disturbances, flow velocity and geometric considerations. For example, the critical Reynolds number for the boundary layer flow over a flat plate reaches about 500,000 based on the free stream velocity outside of the boundary layer and characteristic length of the distance from the leading edge of the plate.