What Is the Reynolds Number?
The Reynolds number (Re) is a dimensionless quantity used in fluid mechanics to predict whether fluid flow will be laminar or turbulent. It represents the ratio of inertial forces to viscous forces in a fluid in relative motion. A high Reynolds number means inertial forces dominate and the flow tends to be turbulent; a low Reynolds number means viscous forces dominate and the flow is laminar.
Reynolds number has broad real-world applications: describing water flow in a pipe, airflow around an aircraft wing, or an object moving through a liquid. It is also essential for scaling between small models and full-size systems — keeping Re the same ensures dynamically similar behaviour.
Reynolds Number Units
The Reynolds number is dimensionless — it has no units. The formula is written so that all physical units cancel out, leaving a pure number that characterizes the flow regime.
Laminar vs Turbulent Flow
Our Reynolds number calculator classifies flows into three regimes:
- Laminar flow (Re < 2 100): Smooth, orderly layers of fluid sliding past each other. Viscous forces dominate. The classic example is slow, syrup-like flow in a narrow tube.
- Transitional flow (2 100 ≤ Re ≤ 3 000): The flow alternates erratically between laminar and turbulent. This regime is unstable and sensitive to external disturbances.
- Turbulent flow (Re > 3 000): Chaotic eddies, vortices, and mixing dominate. Inertial forces rule. Most everyday pipe flows and natural wind patterns are turbulent.
These thresholds (2 100 / 3 000) are specific to pipe flow. For flow over flat plates, around spheres, or in open channels the critical Reynolds numbers differ, but the same physical principles apply.
How to Calculate Reynolds Number — The Formula
The Reynolds number calculator uses two equivalent equations depending on what fluid properties you know:
Re = ρ × u × L / μ (using dynamic viscosity)
Re = u × L / ν (using kinematic viscosity)
where:
- Re — Reynolds number (dimensionless)
- ρ — Fluid density (kg/m³ or lb/ft³)
- u — Fluid velocity relative to the object (m/s or ft/s)
- L — Characteristic linear dimension (m or ft) — e.g., pipe diameter, plate length, sphere radius
- μ — Dynamic (absolute) viscosity of the fluid (mPa·s or lb/(ft·s))
- ν = μ / ρ — Kinematic viscosity (mm²/s or ft²/s)
Characteristic Length
The characteristic length L depends on the geometry of the problem:
- Pipe flow: use the internal diameter of the pipe
- Flow over a flat plate: use the length of the plate along the flow direction
- Flow around a sphere or cylinder: use the diameter
- Airfoil / wing: use the chord length
Built-in Fluid Presets
For convenience, the calculator includes common fluid presets with their standard properties:
| Fluid | ρ (kg/m³) | μ (mPa·s) | ν (mm²/s) |
|---|---|---|---|
| Water at 20 °C (68 °F) | 998.2 | 1.002 | 1.004 |
| Water at 10 °C (50 °F) | 999.7 | 1.307 | 1.307 |
| Air at 20 °C (68 °F) | 1.204 | 0.0181 | 15.1 |
| Engine oil at 40 °C (104 °F) | 876 | 90 | 103 |
You can also enter custom values for any fluid.
Example Calculation
Let's find the Reynolds number for tap water (10 °C) flowing through a pipe of diameter 2.5 cm at 1.7 m/s — a typical household cold-water supply.
- ρ = 999.7 kg/m³
- u = 1.7 m/s
- L = 0.025 m (pipe diameter)
- μ = 1.307 mPa·s = 0.001307 Pa·s
Re = 999.7 × 1.7 × 0.025 / 0.001307 ≈ 32 483
Re ≈ 32 483, which is well above 3 000, so the flow is turbulent. This means the water mixes vigorously inside the pipe — exactly what we observe in everyday plumbing.
Frequently Asked Questions
What Reynolds number is laminar?
For pipe flow, Re < 2 100 is considered laminar. In other geometries the threshold varies.
What Reynolds number is turbulent?
For pipe flow, Re > 3 000 is fully turbulent. Between 2 100 and 3 000 is the transitional regime.
What is kinematic vs dynamic viscosity?
Dynamic viscosity (μ) measures the internal resistance to flow. Kinematic viscosity (ν = μ / ρ) is the dynamic viscosity divided by fluid density — it represents how quickly a fluid diffuses momentum. Kinematic viscosity is often tabulated for common fluids since it directly enters many flow equations.
Is Reynolds number always dimensionless?
Yes. As long as consistent units are used (all SI or all imperial), the units cancel and Re is a pure dimensionless number. Our calculator handles both metric (SI) and American (imperial) unit systems.