What Is the Reynolds Number?

The Reynolds number (Re) is a dimensionless number related to fluid mechanics. It is among the most important attributes used for summarizing the forces acting on a fluid and, based on its value, the turbulence or lack of turbulence of a fluid is determined. The designation is named for Osborne Reynolds, who made many pioneering studies in fluid mechanics in the late 19th and early 20th centuries. The variations in the quantity are laid out on the X-axis of the Moody Chart, one of the more useful graphs in fluid mechanics.

More specifically, the Reynolds number is defined as the ratio of inertial forces, which contribute to turbulence, to viscous forces, which act against turbulence, within a fluid. Put another way, the number describes how likely flow is to be laminar or turbulent for a given set of physical conditions. Laminar, or smooth, flow indicates that everything in the flow of a fluid is moving in the same direction and these internal flows do not affect one another. Turbulent flow, on the other hand, indicates that disruptions or eddies are created within the main flow.

The most common example of laminar and turbulent flow can be found at a sink. When the water is first turned on and is not flowing very fast, it is clear. Most of the internal flows of the water do not interact with one another and move in the same direction; this is laminar flow and indicates a low Reynolds number. As the amount and speed of the water increases, it turns white. The internal flows begin to collide with one another in a turbulent flow, introducing air into the water stream.

Another example of the concept is to imagine an object moving through a fluid. The faster the object moves, the denser the liquid, and the more time the object moves, the more likely the fluid flow is to be turbulent. The more viscous or sticky a fluid is, the greater the chance the fluid’s thickness will act against a turbulent flow.

Mathematically, the Reynolds number is defined as:

Re = ρ * V * L / µ
Where Re = Reynolds number
ρ = fluid density (usually lb/ft³ or 3)
V = velocity (usually ft/s or m/s)
L = length of travel (usually ft or m)
In a pipe or channel, L = hydraulic radius (usually ft or m)
µ = fluid dynamic viscosity (usually lb/(ft*s) or kg/(m*s) or Pa*s)

From the equation, it can be seen that the Reynolds number is in direct proportion to the length. It also varies proportionally to the length and the fluid density. The numbers ρ, V and L all contribute to the inertial forces, whereas µ contributes only to the viscous forces.

For Re of 2,300 or less, fluid flow is considered to be laminar. Turbulent flow, on the other hand, is achieved when Re is greater than 4,000. Values for the Reynolds number between these two quantities indicate transitional flows, which can exhibit characteristics of both types of flow.

The Reynolds number is used in many different applications of fluid mechanics. It is a necessary part of friction factor calculations in some equations in fluid mechanics, such as the Darcy–Weisbach equation. Another common use of the number comes in the modeling of organisms swimming through water, and this application has been done from the largest animals — such as the blue whale — to very small animals, including microorganisms. It even has applications in modeling airflow around objects, such as the wings of an aircraft.