Formula & Engineering Reference
| Symbol | Variable | Unit (SI) |
|---|---|---|
| Re | Reynolds number (dimensionless) | — |
| ρ | Fluid density | kg/m³ |
| V | Mean flow velocity | m/s |
| D | Pipe inside diameter | m |
| μ | Dynamic viscosity | Pa·s |
| ν | Kinematic viscosity (μ/ρ) | m²/s |
| ε | Absolute pipe roughness | m |
Laminar (Re < 2300): f = 64 / Re — exact, independent of roughness.
Turbulent (Re > 4000), Swamee-Jain:
The Swamee-Jain equation is an explicit approximation of the implicit Colebrook-White equation and matches the Moody diagram to within about 1% over 5000 < Re < 10⁸ and ε/D up to 0.05.
Water at 20 °C flowing at 2 m/s through NPS 4 SCH 40 pipe (ID = 102.3 mm):
ρ = 998 kg/m³, μ = 1.0 cP = 0.001 Pa·s, D = 0.1023 m, V = 2 m/s.
Re = (998 × 2 × 0.1023) / 0.001 = 204,191 → well above 4000, so the flow is turbulent.
With commercial steel roughness ε = 0.045 mm, relative roughness ε/D = 0.045/102.3 = 0.00044. Swamee-Jain gives f ≈ 0.0186, which agrees with the Moody chart reading at this Re and ε/D. That friction factor then feeds straight into a Darcy-Weisbach pressure-drop calculation.
Mean velocity, not peak. Re uses the bulk (cross-sectional average) velocity V = Q/A. Don't use the centreline velocity, which is roughly twice the mean in laminar flow.
The transition band is fuzzy. Between Re 2300 and 4000 the flow flickers between laminar and turbulent and no friction correlation is reliable. This tool extends the laminar 64/Re relation through the band as a conservative placeholder and flags it — size critical systems on the turbulent assumption if you might land here.
Newtonian fluids only. The single-value viscosity assumption holds for water, light hydrocarbons, air, and most gases. Slurries, polymers, and greases are non-Newtonian and need an apparent viscosity at the actual shear rate.
Properties at operating conditions. Density and viscosity both change with temperature (and density with pressure for gases). Using 20 °C water properties for a 90 °C line will overstate viscosity and understate Re. Always pull properties at the real operating point.
Round, full pipes. The 2300/4000 thresholds are for circular pipes running full. For ducts or open channels, substitute the hydraulic diameter Dh = 4A/P and expect different transition values.
It is the dimensionless ratio of inertial to viscous forces, Re = ρVD/μ. When viscous forces dominate (low Re) the flow stays orderly and laminar; when inertial forces dominate (high Re) small disturbances grow into turbulent eddies. It is the single most useful number for characterising any internal or external flow.
For full circular pipes: laminar below 2300, transitional 2300–4000, turbulent above 4000. These are conventional engineering values — the real transition depends on inlet disturbances, roughness, and vibration, so treat the band as a caution zone rather than a precise switch.
Laminar: f = 64/Re exactly. Turbulent: from the Moody diagram, or the Swamee-Jain explicit equation used here, which approximates Colebrook-White to about 1%. The factor then drives the Darcy-Weisbach pressure-drop equation, hf = f (L/D)(V²/2g).
In laminar flow a thick viscous sublayer covers the wall texture, so the surface roughness never touches the bulk flow and f = 64/Re regardless of finish. Once turbulent, roughness elements poke through the sublayer and disturb the flow, raising the friction factor — which is why ε/D enters only the turbulent correlations.
Enter dynamic (absolute) viscosity in cP (= mPa·s). The calculator derives kinematic viscosity ν = μ/ρ for you. Both forms of Reynolds number are equivalent: Re = ρVD/μ = VD/ν.
Yes — Reynolds number is fluid-agnostic. Use the gas density and viscosity at the operating temperature and pressure. Gases have low density and viscosity, so they usually reach turbulent flow at modest velocities.
Reynolds Number Engineering Guide
3 topics • Flow regime & friction referenceThe Reynolds number is the first thing most engineers reach for when they look at a flow problem. It collapses four variables — density, velocity, diameter, and viscosity — into one dimensionless number that tells you which physics is in charge. Below the transition, viscosity wins and the flow moves in smooth, predictable layers. Above it, inertia wins and the flow becomes a chaotic mix of eddies. Almost every downstream correlation in fluid mechanics, from friction factor to heat-transfer coefficient, is written as a function of Re.
Getting it right matters because the friction factor — and therefore the pressure drop, pump power, and control-valve authority — depends entirely on which regime you are in. A line that runs laminar behaves very differently from one that runs turbulent, and the calculation path forks at the Reynolds number. This guide covers how the number is built, how it maps to the Moody diagram, and where the practical pitfalls hide.