Reynolds Number Calculator

What the Reynolds number calculator does

The Reynolds number tells you whether a flow will be smooth and orderly or chaotic and churning. This calculator works it out from the flow speed, a characteristic length such as a pipe's diameter, and the fluid's properties, which you can give either as density and dynamic viscosity together or as a single kinematic viscosity.

Below is what the Reynolds number means, the equations behind it, and how to read it to judge the flow.

How to use it

1. Choose how to give the fluid properties: as dynamic viscosity with density, or as kinematic viscosity alone.
2. Enter the flow velocity, the characteristic length, and the viscosity values the method needs.
3. Press Calculate for the Reynolds number, or Reset to clear it.

What the Reynolds number is

The Reynolds number is a single, unitless number that compares two competing influences in a moving fluid. One is inertia, the tendency of the fluid to keep moving and to break into swirls; the other is viscosity, the internal friction that smooths motion out and keeps it orderly. The Reynolds number is the ratio of the two, a measure of which one is winning.

This turns out to be one of the most useful numbers in all of fluid mechanics, because the character of a flow depends on that balance rather than on any single quantity. When viscosity dominates, the flow is smooth; when inertia dominates, it becomes turbulent. The Reynolds number captures the tipping point between them, and because it is dimensionless, the same value means the same kind of flow whether you are looking at water in a pipe, air over a wing, or oil in a bearing.

The equations it uses

In its usual form, the Reynolds number combines the fluid density ρ, the flow velocity v, a characteristic length D, and the dynamic viscosity μ:

Re = ρ v D ÷ μ

Often the density and viscosity are bundled together into the kinematic viscosity, written ν, which is the dynamic viscosity divided by the density. In that case the formula is simpler, and the calculator offers it as the second method:

Re = v D ÷ ν

Both give the same number; they are just two ways of describing the fluid's resistance to flow. The characteristic length is whatever size best describes the flow, most often the diameter of the pipe the fluid runs through.

Laminar, turbulent, and the transition

The value of the Reynolds number sorts flows into types. At low values, below about 2,300 for flow in a pipe, the flow is laminar: the fluid moves in smooth, parallel layers that slide past one another without mixing, like a steady stream from a gently opened tap. Viscosity is in control, and the flow is calm and predictable.

At high values, above roughly 4,000, the flow is turbulent: it breaks into eddies and swirls that mix the fluid thoroughly, like the churning from a tap opened wide. Inertia dominates, and the motion is irregular. Between those thresholds lies a transitional range, where the flow can switch between the two. This matters in practice because the two regimes behave very differently, and many flow methods assume one or the other: smooth-flow formulas like Poiseuille's law expect laminar flow, while pipe-friction formulas for water assume turbulent flow. The Reynolds number tells you which world you are in.

Units and precision

The calculator works in SI units: velocity in metres per second, the characteristic length in metres, density in kilograms per cubic metre, dynamic viscosity in pascal-seconds, and kinematic viscosity in square metres per second. The Reynolds number itself has no units, since it is a ratio, so the same value carries the same meaning across any fluid or scale. Results are shown to several decimal places.

A worked example

Take water flowing at 2 metres per second through a pipe 0.05 metres across, with water's density of about 1000 kg/m³ and dynamic viscosity of about 0.001 pascal-seconds.

The Reynolds number is ρvD ÷ μ = (1000 × 2 × 0.05) ÷ 0.001 = 100,000. That is far above the turbulent threshold, so this flow is firmly turbulent, churning and well mixed, as fast water in a pipe of this size usually is.

Questions people ask

What is the Reynolds number?

It is a unitless ratio of inertial to viscous forces in a fluid, used to predict whether a flow will be smooth (laminar) or chaotic (turbulent). It is one of the central numbers in fluid mechanics.

How do you calculate it?

Multiply density, velocity, and a characteristic length, then divide by the dynamic viscosity, Re = ρvD/μ. Equivalently, divide velocity times length by the kinematic viscosity.

What is the difference between laminar and turbulent flow?

Laminar flow is smooth, with the fluid moving in orderly parallel layers; turbulent flow is chaotic, breaking into mixing eddies. Low Reynolds numbers give laminar flow, high ones give turbulent.

What Reynolds number means turbulent flow?

For flow in a pipe, roughly below 2,300 is laminar and above about 4,000 is turbulent, with a transitional range in between where the flow can shift between the two.

References

A quick note on where the physics comes from. The Reynolds number as the ratio of inertial to viscous forces, and its thresholds for laminar and turbulent flow, are standard fluid mechanics, set out in OpenStax's University Physics and described in the Wikipedia article on the Reynolds number. The SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 14.7, Viscosity and Turbulence. https://openstax.org/books/university-physics-volume-1/pages/14-7-viscosity-and-turbulence
2. Wikipedia, Reynolds number. https://en.wikipedia.org/wiki/Reynolds_number
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811