What is the Reynolds number?

The Reynolds number (Re) is a dimensionless number that predicts whether fluid flow will be laminar or turbulent. It is calculated as Re = rho times v times D divided by mu, where rho is the fluid density in kg per cubic metre, v is the flow velocity in m per second, D is the pipe diameter (or hydraulic diameter) in metres, and mu is the dynamic viscosity in pascal-seconds. A low Re indicates smooth, orderly laminar flow; a high Re indicates chaotic turbulent flow.

What Reynolds number separates laminar from turbulent flow?

For flow in a circular pipe, laminar flow typically exists when Re is below about 2300. Turbulent flow is fully established above about 4000. The range between 2300 and 4000 is called the transitional regime, where flow can switch between laminar and turbulent unpredictably. These thresholds can vary with pipe roughness, inlet conditions, and other factors.

Why does the Reynolds number matter in engineering?

The flow regime affects friction losses, heat transfer efficiency, and mixing. Laminar flow has low friction losses and predictable velocity profiles, making it useful in microfluidics and lubrication. Turbulent flow mixes fluids efficiently and improves heat transfer, which is why turbulence is often desirable in heat exchangers. Knowing the Reynolds number lets engineers choose appropriate pipe sizes, flow rates, and fluid properties to achieve the desired regime.

Reynolds Number Calculator

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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Fluid preset

Fluid density ρ

kg/m³

Dynamic viscosity μ

Pa·s

Flow velocity v

m/s

Pipe diameter D

100,000

Reynolds number (Re)

Flow regimeTurbulent

Critical v (Re=2300)0.046 m/s

Turbulent v (Re=4000)0.080 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 v_crit = 2300 × μ / (ρ × D). For fully turbulent flow: v_turb = 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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