Reynolds Number Calculator
Find the Reynolds number and the flow regime — laminar, transitional, or turbulent — from fluid and flow properties.
Re > 4000 — chaotic mixing and eddies; inertial forces dominate. Re is dimensionless. Thresholds are the standard pipe-flow convention: laminar Re < 2300, transitional 2300–4000, turbulent Re > 4000.
The Reynolds number is Re = ρvD ÷ μ, where ρ is fluid density, v is velocity, D is the pipe diameter, and μ is dynamic viscosity. Water (ρ = 998 kg/m³, μ = 0.001002 Pa·s) at 2 m/s in a 0.05 m pipe gives Re = 998 × 2 × 0.05 ÷ 0.001002 ≈ 99,600 — turbulent, since Re > 4000.
What the Reynolds number means
The Reynolds number (Re) is a dimensionless ratio that compares the two forces competing in a moving fluid: the inertial forces that carry the fluid forward (ρvD) against the viscous forces that resist internal shearing (μ). A small Re means viscosity wins — the flow moves in smooth, parallel layers. A large Re means inertia wins — the flow breaks into chaotic eddies and mixing. Because it is a pure ratio, the same Re predicts the same qualitative flow pattern whether you are studying blood in a capillary, oil in a pipeline, or air over a wing.
Re = Reynolds number (dimensionless), ρ = density (kg/m³), v = velocity (m/s), D = characteristic length or pipe diameter (m), μ = dynamic viscosity (Pa·s)
Worked example
Water at 20 °C (ρ = 998 kg/m³, μ = 0.001002 Pa·s) flows at 2 m/s through a pipe of 0.05 m internal diameter. Find the Reynolds number and classify the flow.
- 1 Collect the four properties in SI units. ρ = 998 kg/m³, v = 2 m/s, D = 0.05 m, μ = 0.001002 Pa·s. Keeping every value in base SI units makes Re come out dimensionless.
- 2 Multiply the numerator ρ·v·D. 998 × 2 × 0.05 = 99.8 kg/(m·s). These are the inertial terms of the flow.
- 3 Divide by the dynamic viscosity μ. Re = 99.8 ÷ 0.001002 ≈ 99,600. The units cancel completely, leaving a pure number.
- 4 Classify the flow regime. 99,600 is far above 4000, so the flow is turbulent. Laminar is Re < 2300; transitional is 2300–4000.
Pipe-flow regimes and example fluids
Regime thresholds follow the standard pipe-flow convention. Fluid properties are quoted at about 20 °C.
| Item | Value | Notes |
|---|---|---|
| Laminar | Re < 2300 | Smooth, orderly layers; viscosity dominates |
| Transitional | 2300 ≤ Re ≤ 4000 | Unstable, intermittent bursts of turbulence |
| Turbulent | Re > 4000 | Chaotic eddies and mixing; inertia dominates |
| Water (20 °C) | ρ = 998 kg/m³, μ = 0.001002 Pa·s | Common reference liquid |
| Air (20 °C) | ρ = 1.204 kg/m³, μ = 1.81×10⁻⁵ Pa·s | Low density and viscosity |
| Olive oil (20 °C) | ρ = 915 kg/m³, μ = 0.081 Pa·s | High viscosity keeps Re low |
Choosing the characteristic length and reading the result
The characteristic length D is whatever length scale defines the flow geometry. For a round pipe it is the internal diameter; for flow over a flat plate it is the distance along the plate; for a non-circular duct it is the hydraulic diameter (4 × area ÷ wetted perimeter). Pick the wrong length and the number is meaningless, so always state which D you used.
The 2300 and 4000 thresholds are the standard convention for flow inside a pipe and are not universal — external flows and boundary layers transition at very different values, often near Re ≈ 5×10⁵ for a flat plate. The transitional band between 2300 and 4000 is genuinely unsettled: real pipes can hold laminar flow well past 2300 if disturbances are minimal, or trip early if the inlet is rough. Treat a transitional result as a warning that the flow is sensitive rather than a precise prediction.