Perhaps the most "famous" dimensionless group, but by no means the most "important", is the Reynolds number.

The **Reynolds number** is a dimensionless group that relates
the inertial forces to the viscous forces, and is written as:

$\displaystyle{Re = \frac{\textstyle{inertial force}}{\textstyle{viscous force}} = \frac{UD\rho}{\mu} = \frac{UD}{\nu}}$

The most important function of the Reynolds number, not surprisingly, is to determine when inertial forces are bigger/smaller than viscous forces (which translates into whether we have a Stoke's flow (viscous-dominated flow) or not).

Perhaps the most employed function of the Reynolds number, however, is as an empirical measure of whether a flow is laminar or turbulent. Let's take a pipe flow as an example:

In a pipe flow, the velocity profile varies depending on whether the flow is laminar or turbulent:

We get the flow on the left if Re<2300 and the flow on the right if Re>5000. (Where did these numbers come from?! What happens at Re=3400?)

It is *critical* to note that these values of Re are
empirical and refer *only* to flow in a pipe. Other
transition values (for different geometries) are equally obscure
and must be looked up (for example, flow past a sphere becomes
turbulent at Re=340,000!)

Calculate the Reynolds Number and use it to predict flow regimes