Reynolds Number Calculator

How to use the Reynolds number calculator

Start with the Solve via buttons. Choose dynamic viscosity and density if you have those two properties of the fluid, or kinematic viscosity if you have that single figure instead. Then enter the flow velocity and the characteristic length. Every input has a unit menu, so you can mix, for example, a speed in ft/s with a length in millimetres.

• Flow velocity: the speed of the fluid relative to the pipe wall or the object, in units from metres per second to knots.
• Characteristic length: the length that sets the scale of the flow. By convention this is the inner diameter for flow in a pipe, the diameter for a sphere and the distance from the leading edge for a flat plate.
• Dynamic viscosity and density or kinematic viscosity describe the fluid. All of the fluid fields for the chosen mode must be filled in.

The result is a single number with no unit. A zero length gives a Reynolds number of 0.

What is the Reynolds number?

The Reynolds number (Re) compares two effects in a moving fluid: inertia, which keeps the fluid going and tends to stir it into eddies, and viscosity, which damps out disturbances and keeps neighbouring layers moving together. When viscosity wins, Re is small and the flow is laminar, with smooth parallel streamlines. When inertia wins, Re is large and the flow turns turbulent, with chaotic swirling.

Because it is dimensionless, two flows with the same Reynolds number and the same geometry behave alike even if their sizes and fluids differ. That is why engineers can test a scale model of an aircraft wing in a wind tunnel or water channel and carry the results over to the real thing.

Treat these limits as approximate. A very smooth pipe with a calm inlet can stay laminar well past 2,300, and rough walls or vibration can trigger turbulence earlier. Other geometries have their own critical values; along a smooth flat plate the boundary layer typically becomes turbulent near Re ≈ 500,000, measured from the leading edge.

Reynolds number formula

Re = ρ × v × L ÷ μρ = density, v = velocity, L = characteristic length, μ = dynamic viscosity

Dividing the dynamic viscosity by the density gives the kinematic viscosity ν = μ ÷ ρ, which leads to the second form used by the calculator:

Re = v × L ÷ ν

In SI units the dynamic viscosity is measured in pascal-seconds, and 1 Pa·s equals 1 N·s/m² and 1 kg/(m·s). The older CGS unit is the poise, equal to 0.1 Pa·s, and the centipoise is one hundredth of a poise. Kinematic viscosity is measured in m²/s; the stokes is 10⁻⁴ m²/s and the centistokes 10⁻⁶ m²/s, which equals 1 mm²/s. Water at about 20 °C has a kinematic viscosity close to 1 cSt.

Worked examples

Water in a pipe. Water at 20 °C (ρ ≈ 998 kg/m³, μ ≈ 0.001 Pa·s) flows at 1.5 m/s through a pipe with a 25 mm inner diameter. Re = 998 × 1.5 × 0.025 ÷ 0.001 = 37,425, well inside the turbulent range.

Air over a plate. Air with ν ≈ 1.5 × 10⁻⁵ m²/s moves at 10 m/s along a plate 0.5 m long. Re = 10 × 0.5 ÷ 1.5 × 10⁻⁵ ≈ 333,333. That is below the typical plate transition value, so the boundary layer at the trailing edge is probably still laminar.

Thick oil. A lubricating oil (ρ ≈ 870 kg/m³, μ = 0.2 Pa·s) flows at 0.5 m/s in a 20 mm pipe. Re = 870 × 0.5 × 0.02 ÷ 0.2 = 43.5, which is solidly laminar, and the flow rate calculator in pressure-difference mode is then a good fit.

The critical speed for water. For Re to reach 2,300 in that 25 mm pipe, the water would need to move at only 2,300 × 0.001 ÷ (998 × 0.025) ≈ 0.092 m/s, about 9 cm per second. Anything faster is above the laminar limit, which is why water in ordinary plumbing is almost always turbulent.

Choosing the characteristic length

The Reynolds number is only meaningful together with the length it was built from, so the choice must match the critical values you compare against.

• Circular pipe: the inner diameter.
• Non-circular duct: the hydraulic diameter, four times the flow area divided by the wetted perimeter.
• Sphere or cylinder in a cross-flow: the diameter.
• Flat plate or wing: the distance along the surface, or the chord length of the wing.

If you need the fluid's density first, use the density calculator. Speeds in other units can be checked with the m/s to mph converter. More tools are on the physics calculators page.