The Reynolds number, abbreviated Re, is a dimensionless number that characterizes the extent of flow turbulence. Turbulence describes the behavior where adjacent layers of fluid mix together. A flow in which lots of mixing occurs between layers is called a turbulent flow. A flow in which little or no mixing occurs between layers is called a laminar flow. The illustrations below show the pathlines (i.e. the trajectories of individual fluid particles) of turbulent and laminar flows. The edge of the pipe causes friction which slows down the fluid. In turbulent flow, mixing between adjacent fluid layers causes an averaging of velocities so that the flow has a uniform velocity profile. In laminar flows there is little or no mixing and so a parabolic flow profile arises, where flow velocity is highest towards the center and lowest towards the edges of the pipe.

## Mean velocity profile

##### Low Reynolds number (laminar flow)

The Reynolds number is an extremely important parameter in flow measurement applications because many types of flow sensors are only suitable for measuring high Reynolds number (i.e. turbulent) flows. The reason for this is that the flow sensors require a uniform velocity profile for reasonable accuracy to be achieved. The Reynolds number is not a material property because although it is highly dependent on fluid viscosity, it also depends on the flow velocity and characteristic length scales. The Reynolds number for the flow of a gas or liquid in a pipe of circular, square or rectangular cross section is defined as:

Here, ρ (kg/m3) is the fluid density, u (m/s) the velocity, D (m) the length scale and µ (Pa·s) the dynamic viscosity. For a pipe or circular cross section, D is the inner diameter. For a pipe of square cross section, D is the inner length of one side. For a pipe of rectangle cross section, D is four times the area, divided by the perimeter. An intuitive approach to grasping the concept of the Reynolds number is to understand it as the ratio of momentum forces to viscous forces. By inspecting the equation, we can observe that Reynolds number increases with increasing flow velocity because of the increase in momentum. Conversely, the Reynolds number decreases with increasing viscosity because of the increase in viscous forces. The Reynolds number decreases with increasing pipe diameter because the flow gradient transverse to the flow direction becomes smaller, thereby reducing the viscous forces. In General, Laminar and turbulent flow regimes are defined by the following Reynolds numbers:

Re < 2,000 for laminar flow

Re > 4,000 for turbulent flow

The flow regime between a Reynolds number of 2,000 and 4,000 is known as the laminar-turbulent transition region and is neither completely laminar nor completely turbulent.