Reynolds' Number

Whether a given flow turns out to be streamline or turbulent depends, as seen in Section 9.2, on the flow speed -- but it also depends on the fluid's density and viscosity, and on the size of the tube or channel the fluid is flowing through. Osborne Reynolds combined all four of these quantities into a single dimensionless number, now called the Reynolds number, that is used to predict which kind of flow will actually occur:

where $\rho$ is the density of the fluid, $v$ is its (average) flow speed, $d$ is a characteristic length of the flow -- typically the diameter of the pipe or tube the fluid is flowing through -- and $\eta$ is the coefficient of viscosity of the fluid.

Why it is dimensionless. Checking the dimensions of each quantity: $[\rho] = \text{kg/m}^3$, $[v] = \text{m/s}$, $[d] = \text{m}$, and $[\eta] = \text{Pa·s} = \text{kg/(m·s)}$. Substituting,

so the Reynolds number carries no unit at all -- it is a pure number, which is exactly what makes it so useful: its numerical value can be compared directly across completely different fluids, tube sizes, and flow speeds, without worrying about the units each individual quantity happens to be measured in.

Physical meaning. The Reynolds number can be understood, physically, as (roughly) a ratio of the inertial forces within the flowing fluid (associated with the fluid's own momentum and its tendency to keep moving in whatever direction it is already going) to the viscous forces within it (associated with the fluid's internal friction, which acts to damp out any disturbance and keep the flow smooth and orderly). When $Re$ is small, viscous forces dominate over inertial ones -- any small disturbance in the flow is quickly damped out by the fluid's own internal friction, and the flow stays smooth and streamline. When $Re$ is large, inertial forces dominate -- a small disturbance is no longer damped out fast enough by viscosity, and instead grows and cascades into the chaotic swirling and mixing of turbulent flow.

Predicting the flow regime. Experimentally, it is found that for flow through a pipe:

• $Re$ below roughly $1000$-$2000$: the flow is streamline (laminar).
• $Re$ above roughly $2000$-$3000$: the flow is turbulent.
• $Re$ in between these two ranges: the flow is unstable, and can switch between streamline and turbulent behaviour. …