The Reynolds number is one of the most important dimensionless numbers in fluid mechanics. It compares inertial forces to viscous forces in a flowing fluid and tells you whether flow will be smooth and ordered (laminar) or chaotic and mixing (turbulent). The formula is Re equals the fluid density multiplied by the flow velocity and the pipe diameter, divided by the dynamic viscosity: Re = rho times v times D divided by mu. For flow in a circular pipe, laminar flow occurs when Re is below about 2300, turbulent flow is established above about 4000, and the range in between is a transitional zone where both types can occur. These thresholds can vary depending on pipe roughness, entrance conditions, and upstream disturbances. The flow regime matters because it determines friction losses, heat transfer rates, and mixing efficiency. Turbulent flow increases friction (and therefore pumping costs) but also enhances heat and mass transfer, which is desirable in heat exchangers and reactors. Laminar flow is preferred in microfluidics, blood flow analysis, and precision coating processes. This calculator accepts fluid density, flow velocity, pipe diameter, and dynamic viscosity, and returns the Reynolds number, the flow regime, the critical velocity at which turbulence begins (Re = 2300), and the velocity at which flow becomes fully turbulent (Re = 4000). Common fluid presets are included for convenience.
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kg/m³
Pa·s
m/s
m
100,000
Reynolds number (Re)
Flow regimeTurbulent
Critical v (Re=2300)0.046 m/s
Turbulent v (Re=4000)0.08 m/s
Kinematic viscosity1.000e-6 m²/s
Critical Re thresholds (2300 / 4000) apply to smooth circular pipes. Roughness, entrance conditions, and non-circular geometries shift these values.
How it works
The Reynolds number is Re = ρ × v × D / μ, where ρ is density in kg/m³, v is velocity in m/s, D is pipe diameter in m, and μ is dynamic viscosity in Pa·s. Kinematic viscosity ν = μ / ρ in m²/s. The critical velocity for laminar-to-turbulent transition is vcrit = 2300 × μ / (ρ × D). For fully turbulent flow: vturb = 4000 × μ / (ρ × D). Regime boundaries: Re < 2300 = laminar; 2300 to 4000 = transitional; Re > 4000 = turbulent.
Worked example
Water at 20 °C (density ρ = 1000 kg/m³, viscosity μ = 0.001 Pa·s) flows at v = 2 m/s through a pipe of diameter D = 0.05 m. Re = 1000 × 2 × 0.05 / 0.001 = 100,000. This is well above 4000, so flow is turbulent. The critical velocity is 2300 × 0.001 / (1000 × 0.05) = 0.046 m/s. These match the default values pre-filled above.
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