Reynolds Number
The Reynolds number (Re) is a dimensionless quantity that predicts the flow regime of a fluid by comparing inertial forces to viscous forces within the flow. A low Reynolds number indicates that viscous forces dominate, resulting in smooth laminar flow, while a high value signals that inertial forces dominate, leading to turbulent flow. It is indispensable in scaling model experiments to full-size systems, designing pipelines, and predicting aerodynamic behaviour around aircraft and vehicles.
Key Formula
Re = (ρ × v × L) / μ = (v × L) / ν
LaTeX: Re = \frac{\rho v L}{\mu} = \frac{v L}{\nu}
| Symbol | Meaning | Unit |
|---|---|---|
| Re | Reynolds number | dimensionless |
| ρ | Fluid density | kg/m³ |
| v | Characteristic flow speed | m/s |
| L | Characteristic length (e.g., pipe diameter) | m |
| μ | Dynamic viscosity | Pa·s |
| ν | Kinematic viscosity (ν = μ/ρ) | m²/s |
Worked Example
Problem
Water at 20 °C (ρ = 998 kg/m³, μ = 1.002 × 10⁻³ Pa·s) flows at 1.5 m/s through a pipe of diameter D = 0.05 m. Determine the Reynolds number and identify the flow regime.
Solution
Step 1 — Use Re = ρvD/μ. Step 2 — Substitute: Re = (998 × 1.5 × 0.05) / (1.002 × 10⁻³). Step 3 — Numerator: 998 × 1.5 × 0.05 = 74.85. Step 4 — Re = 74.85 / (1.002 × 10⁻³) ≈ 74 700. Step 5 — Since Re > 4000, the flow is turbulent.
Answer
Re ≈ 74 700 — Turbulent flow
Reynolds Number Ranges and Corresponding Flow Regimes (Pipe Flow)
| Re Range | Flow Regime | Velocity Profile | Typical Example |
|---|---|---|---|
| Re < 2300 | Laminar | Parabolic | Blood in capillaries, oil in narrow tubes |
| 2300 ≤ Re ≤ 4000 | Transitional | Unstable, intermittent | Slowly flowing river near bank |
| Re > 4000 | Turbulent | Nearly flat (plug) | Water in household pipes |
| Re > 10⁵ | Fully turbulent | Flat with thin boundary layer | Industrial cooling water mains |
| Re ~ 10⁶ | Highly turbulent | Near-plug with thin laminar sublayer | Aircraft wake, ship hulls |
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Related Terms
Laminar Flow
Laminar flow is a smooth, orderly regime of fluid motion in which fluid particles travel in parallel layers (laminae) without lateral mixing or cross-current fluctuations. It occurs at low Reynolds numbers (typically Re < 2300 in pipes) where viscous forces dominate over inertial forces, producing a parabolic velocity profile in pipe flow. Laminar flow is essential in microfluidics, blood flow in capillaries, lubrication engineering, and precision chemical dosing.
Turbulent Flow
Turbulent flow is a chaotic, irregular fluid motion characterised by rapid fluctuations in velocity and pressure, eddies, vortices, and vigorous lateral mixing between fluid layers. It occurs when inertial forces overcome viscous forces, typically at Reynolds numbers above 4000 in pipe flow, and is the dominant regime in most industrial, atmospheric, and oceanic flows. Despite its complexity, turbulent flow enhances heat and mass transfer, making it beneficial in heat exchangers and combustion systems.
Viscosity
Viscosity is a measure of a fluid's resistance to deformation or flow under an applied shear stress, arising from internal friction between adjacent fluid layers moving at different velocities. Dynamic (absolute) viscosity quantifies the shear stress needed to produce a unit velocity gradient, while kinematic viscosity is the ratio of dynamic viscosity to fluid density. Viscosity governs flow behaviour in lubrication, blood circulation, polymer processing, and aerodynamics.
Named after Osborne Reynolds (1842–1912), a British engineer who introduced this dimensionless group in his seminal 1883 paper on the conditions for laminar and turbulent flow in pipes. The term "Reynolds number" was later coined by Arnold Sommerfeld in 1908.