Reynolds number calculator

What is the Reynolds number?

A dimensionless number that tells whether flow in a pipe is smooth or turbulent: Re = ρ v D ÷ μ, where ρ is the fluid density, v its average speed, D the inside diameter and μ the dynamic viscosity. OpenStax University Physics writes the same thing with the radius, N_R = 2ρvr ÷ η. With kinematic viscosity ν = μ ÷ ρ it is simply Re = v D ÷ ν.

When is flow laminar or turbulent?

For flow in a tube, OpenStax gives: below about 2,000 the flow is laminar; above about 3,000 it is turbulent; in between it is unstable and can flip between the two, depending on obstructions and surface roughness.

Example: air in a duct

In an OpenStax example, air (ρ = 1.23 kg/m³, η = 1.81 × 10⁻⁵ Pa·s) moves at 0.15 m/s through a duct of radius 9 cm (D = 0.18 m): Re = 1.23 × 0.15 × 0.18 ÷ 1.81 × 10⁻⁵ ≈ 1,835, so the flow is laminar. It stays laminar up to about 0.16 m/s. Choose “Other fluid” to enter those properties.

Where do the water and air values come from?

Water density and viscosity at 1 atm are from the NIST Chemistry WebBook, which uses the international IAPWS formulations — for example 998.21 kg/m³ and 1.0016 mPa·s at 20 °C (68 °F). Air is the U.S. Standard Atmosphere, 1976 at sea level: 1.225 kg/m³ and 1.7894 × 10⁻⁵ Pa·s at 15 °C (59 °F).

Why does warm water become turbulent sooner?

Its viscosity drops fast with temperature: 1.31 mPa·s at 10 °C but only 0.354 mPa·s at 80 °C (NIST), while density barely changes. At the same speed and pipe size, 80 °C water has a Reynolds number about 3.6 times that of 10 °C water.

Is the Reynolds number the same for any shape of pipe?

The formula uses a single length. For a round pipe it is the inside diameter. For ducts that are not round, engineers use the hydraulic diameter, four times the flow area divided by the wetted perimeter, and the transition values then differ somewhat.

What the number compares

The Reynolds number is the ratio of inertial forces, which tend to make a flow chaotic, to viscous forces, which smooth it. It can be written with dynamic viscosity μ as ρvD ÷ μ, or with the kinematic viscosity ν = μ ÷ ρ as vD ÷ ν. Because it has no units, flows of different fluids and sizes can be compared directly. Doubling the speed or the diameter doubles the Reynolds number, and warming water lowers its viscosity and raises the number.

Worked examples

Water at 20 °C (ρ = 998.21 kg/m³, μ = 1.0016 mPa·s) moving at 1 m/s through a 1 inch (0.0254 m) pipe has Re = 25,314, which is turbulent. To keep that pipe laminar, the speed would have to be below about 0.079 m/s, where Re reaches 2,000; at 0.05 m/s the Reynolds number is 1,266 (laminar). Sea-level air (ρ = 1.225 kg/m³, μ = 1.7894 × 10⁻⁵ Pa·s) at 5 m/s in a 0.3 m duct has Re = 102,688, which is turbulent.

Where the thresholds come from

The calculator uses the OpenStax figures for flow in a tube: laminar below about 2,000 and turbulent above about 3,000, with an unstable range in between. Other textbooks draw the lines at slightly different values, commonly 2,300 for the start of transition and 4,000 for turbulence, because the actual change depends on how smooth the pipe is and how disturbed the flow is at the inlet. Treat the boundaries as a band, not as a switch.

Common mistakes

Using the radius where the diameter is needed halves the answer; the OpenStax form 2ρvr ÷ η has the extra 2 for that reason. Mixing the two viscosities is another: dynamic viscosity (Pa·s) goes with density, kinematic viscosity (m²/s) is used on its own. Unit slips with centipoise (1 cP = 1 mPa·s) and centistokes (1 cSt = 1 mm²/s) are the third. Finally, take the viscosity at the actual fluid temperature, because for water it changes by a factor of nearly four between 10 and 80 °C.

Sources: OpenStax University Physics Volume 1, section 14.7 (Viscosity and Turbulence); water properties from the NIST Chemistry WebBook (IAPWS formulations) and air from the U.S. Standard Atmosphere, 1976; unit conversions from NIST SP 811. Related: pipe velocity calculator, friction loss calculator, Bernoulli equation calculator and pipe size calculator.