Reynolds Number Calculator

How to use the Reynolds number calculator

Enter the average flow velocity and the pipe's inner diameter, each in whatever unit you have. Use the bore, not the nominal pipe size, which can be noticeably different. If you know the flow rate instead of the velocity, the average velocity is the flow rate divided by the bore area: V = Q ÷ (π × D² ÷ 4).

Then describe the fluid. Choose Water at 20 °C or Air at 20 °C and 1 atm to use built-in property values, or leave Other fluid selected and enter a viscosity. With a kinematic unit, such as cSt, mm²/s or m²/s, the density isn't needed. With a dynamic unit, such as cP, mPa·s or Pa·s, the calculator divides by the density you enter, so take it at the same temperature as the viscosity.

The results show the Reynolds number, the flow regime it implies for a circular pipe, and the kinematic viscosity that went into the calculation, in cSt.

The Reynolds number formula

Re = V × D ÷ ν = ρ × V × D ÷ μ, where V is the average velocity, D the inner diameter, ν the kinematic viscosity, ρ the density and μ the dynamic viscosity. The two forms are the same because ν = μ ÷ ρ.

Re has no units. It compares the fluid's inertia, which keeps a disturbance going, with its viscosity, which damps the disturbance out. The units only cancel if they are consistent: in SI, meters per second times meters divided by square meters per second leaves a pure number. The calculator converts every input to SI before it divides.

For a duct that isn't round, use the hydraulic diameter in place of D: Dh = 4A ÷ P, where A is the flow area and P is the wetted perimeter. For a square duct, Dh equals the side length.

Worked example: water in a 50 mm pipe

Water at 20 °C flows at 1 m/s through a pipe with a 50 mm bore. NIST gives water's viscosity at that temperature as 1.0016 mPa·s and its density as 998.21 kg/m³. These are the calculator's starting values, and it shows a kinematic viscosity of 1.0034 cSt and a Reynolds number of 49,831, which is turbulent.

By hand: ν = 0.0010016 Pa·s ÷ 998.21 kg/m³ = 1.0034 × 10⁻⁶ m²/s, and Re = 1 m/s × 0.05 m ÷ 1.0034 × 10⁻⁶ m²/s ≈ 49,831. Many textbooks round water's kinematic viscosity to 1.004 cSt. Entering that gives 49,801, less than 0.1% lower.

A hydraulic oil behaves very differently. An ISO VG 46 oil is about 46 cSt at 40 °C. Flowing at 2 m/s in a line with a 20 mm bore, it has a Reynolds number of 870, well inside the laminar range.

In US units, water at 20 °C moving at 6 ft/s through a 2 in bore gives a Reynolds number of 92,588.

Gases turn turbulent at modest speeds too. Air at 20 °C and 1 atm moving at 2 m/s through a 100 mm duct gives 13,280.

Laminar, transitional or turbulent: reading the result

For a full circular pipe, engineers usually call the flow laminar below a Reynolds number of about 2,300, transitional from 2,300 to about 4,000, and turbulent above 4,000. The Engineering ToolBox uses these limits, and so does this calculator. The OpenStax University Physics textbook uses 2,000 and 3,000 instead.

Those limits are conventions, not constants of nature. In a very smooth pipe with a calm, well-rounded inlet, laminar flow can survive well above 4,000. Rough walls, bends, valves, pump pulsation and vibration set off turbulence sooner. Between the limits, the flow can switch back and forth between the two states.

The regime decides how you work out friction. In laminar flow, the Darcy friction factor is exactly 64 ÷ Re. In turbulent flow, it depends on Re and the pipe's relative roughness, and comes from the Colebrook equation or a Moody chart. Neither method is reliable in the transitional range, so designers try to stay out of it.

• The calculator assumes a full pipe with steady flow. Just after an inlet, a bend or a valve, the flow pattern is still developing.
• It is for Newtonian fluids, whose viscosity doesn't change with how fast they are sheared. Slurries, paints and polymer solutions need a modified Reynolds number.
• The 2,300 and 4,000 limits are for pipes only. Flow over a flat plate, a sphere or a wing uses a different length and different transition values.
• For a gas moving faster than about a third of the speed of sound, compressibility also matters, and Re alone doesn't describe the flow.

Typical kinematic viscosities

Viscosity changes strongly with temperature, so use a value at the fluid's actual temperature. These figures are at atmospheric pressure:

• Water at 20 °C: 1.0016 mPa·s and 998.21 kg/m³, so 1.0034 cSt (NIST). Many textbooks round this to 1.002 cP and 1.004 cSt.
• Water at 40 °C: 0.6527 mPa·s and 992.22 kg/m³, so 0.6579 cSt (NIST).
• Air at 20 °C: 18.13 μPa·s and 1.204 kg/m³, so 15.06 cSt (Engineering ToolBox).
• Air at 40 °C: 16.92 cSt (Engineering ToolBox).
• ISO VG 46 hydraulic oil at 40 °C: 41.4 to 50.6 cSt, nominally 46 cSt.