ReynoldsNumber

The Reynolds number (Re) is a dimensionless quantity that is used to predict the flow pattern in fluid dynamics. It is named after Osborne Reynolds who introduced it in the late 19th century.

Reynolds number is a measure of the relative importance of inertial forces to viscous forces in a fluid flow and can be used to predict the onset of turbulence.

The formula for Reynolds number is:

$$Re = \rho uD/\mu = uD/\nu$$

Where:

• $\rho$ is the fluid density
• $u$ is the fluid velocity
• $D$ is the characteristic length (for example, hydraulic diameter for pipe flow)
• $\mu$ is the dynamic viscosity of the fluid
• $\nu$ is the kinematic viscosity of the fluid ($\nu = \mu/\rho$)

For an incompressible viscous fluid flow:

• When Re is low (typically less than 2000 for pipe flow), the flow is typically laminar. In this case, viscous forces dominate, and the fluid particles move in parallel layers (or laminae).
• When Re is high (typically greater than 4000 for pipe flow), the flow is typically turbulent. Here, the inertial forces, which are proportional to fluid velocity, dominate over viscous forces.
• In the range in between, the flow can be transitional, with a mix of laminar and turbulent flow characteristics.

Please note that these ranges for the transition from laminar to turbulent flow are approximations and can depend on other factors, such as the smoothness of the pipe surface and disturbances in the flow.

It's important to note that the Reynolds number isn't only used in the study of pipe flow. It's a general concept that is used in a variety of fluid dynamics applications, including flow over flat plates, around spheres, and in open channels.