Lesson 4 - Principles of Fluid Mechanics 2

When fluid flows down a pipe or a duct, it will flow in one of three so-called regimes (fundamentally different patterns). These regimes depend on the combination of velocity and diameter of the pipe or duct, and also the density and viscosity of the fluid. The three regimes are:

• Laminar

• Turbulent

• Transitional

Most of the time, we do not have the luxury of being able to look inside a pipe to see which pattern (or regime) the fluid has adopted. In practice, we need a numerical indicator. The indicator which we use is termed the Reynolds number (symbol Re), named after Osborne Reynolds who derived a number and observed how this number described the three flow regimes above.

Osborne Reynolds (1842-1912) was an Irish born British mathematician. As well as being a pioneer in fluid mechanics, he may have also been a pioneer in pre-university gap years, as he worked in a shipyard for a year before studying for his university mathematics degree. It is little wonder that his fluid mechanic’s work was as applicable in ship design as it was in pipe flows.

The Reynolds number combines the flow velocity, the internal pipe diameter and the fluid density, and viscosity in the following manner:

Where  is the fluid density, u is the average fluid velocity, d is the internal pipe diameter, and µ is the fluid dynamic viscosity.

Let us do a quick worked example to ensure we can calculate a Reynolds number.

Example:

Bright beer is flowing at 2 m/s down a pipe of 125 mm ID. If the density and viscosity are 1,005 kg/m³ and 2 cP respectively, what is the Reynolds number?

You will notice that there are no units for a Reynolds number – this is not a mistake. The units of the various parameters cancel one another out, which leaves the Reynolds number without a unit. Such numbers are called dimensionless (unit-less) numbers.