How do fluid velocity and pipe diameter affect the Reynolds number?

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differential manometer, Torricellian barometer, and siphon — fluid mechanics Figure 1: Schematic representations of (A) a differential manometer, (B) a Torricellian barometer, and (C) a siphon.

In fluid mechanics, the Reynolds number is a dimensionless quantity used to predict whether flow is laminar or turbulent. It is calculated using the following relationship:

$R e = \frac{ρ v d}{μ}$

Where:
* $ρ$ is the fluid density
* $v$ is the average velocity of the flow
* $d$ is the diameter of the tube
* $μ$ is the absolute viscosity

Because velocity and pipe diameter are both in the numerator of this equation, the Reynolds number is directly proportional to both variables. This means that as either the fluid velocity increases or the pipe diameter increases, the Reynolds number increases. Consequently, higher velocities or larger diameters make the flow more likely to transition from a steady, streamlined laminar state to an unsteady, turbulent state.1