Water in a pipe
Given: ρ = 998 kg/m³, v = 1.5 m/s, D = 0.05 m, μ = 0.001 Pa·s
Working: Re = 998 × 1.5 × 0.05 / 0.001
Re ≈ 74,850, indicating turbulent pipe flow under common criteria.
Fluid Mechanics
Reynolds Number Formula
Learn the Reynolds number formula Re = ρvD/μ, understand each variable, calculate flow regime and use a live calculator for pipe-flow Reynolds number.
Governing equation
Re = ρ × v × D / μ = v × D / ν
Reynolds number is a dimensionless ratio comparing inertial and viscous effects in a flowing fluid. It is widely used to assess whether internal flow is broadly laminar, transitional or turbulent.
For circular pipe flow, D is the internal pipe diameter. Typical textbook thresholds are approximately Re < 2300 laminar, 2300–4000 transitional and Re > 4000 turbulent, although real systems can vary.
Live calculator
Pipe-flow Reynolds number calculator
Uses Re = ρvD/μ with SI inputs.
Result
74,850Re
Approximate pipe-flow regime: turbulent
Formula variables and units
| Symbol | Meaning | Typical SI unit |
|---|---|---|
| Re | Reynolds number | dimensionless |
| ρ | Fluid density | kg/m³ |
| v | Mean flow velocity | m/s |
| D | Characteristic diameter or length | m |
| μ | Dynamic viscosity | Pa·s |
How to calculate it
- 1Use fluid properties at the relevant operating temperature.
- 2Determine mean velocity and characteristic diameter/length.
- 3Keep density, velocity, length and viscosity in compatible SI units.
- 4Calculate Re = ρvD/μ.
- 5Interpret the result using the flow geometry and applicable regime criteria.
Common mistakes
- Using pipe radius instead of diameter when the formula expects D.
- Using viscosity in cP without conversion to Pa·s.
- Using fluid properties at the wrong temperature.
- Applying pipe-flow threshold values blindly to different geometries.
Worked examples
Viscosity effect
Given: Same density, velocity and diameter but viscosity doubles
Working: Re is inversely proportional to μ
Reynolds number is reduced by half.
Go deeper
Use the dedicated OneCalcApp tool
The quick calculator above demonstrates the governing equation. Open the related engineering tool for broader inputs, outputs and calculation context where available.
Frequently asked questions
What is the Reynolds number formula?
A common form is Re = ρvD/μ. Using kinematic viscosity ν, it can also be written Re = vD/ν.
Is Reynolds number dimensionless?
Yes. The units cancel when the equation is used consistently.
What Reynolds number is turbulent in a pipe?
A common engineering rule is that fully developed internal pipe flow above about Re 4000 is turbulent, with a transitional region between roughly 2300 and 4000.
Does temperature affect Reynolds number?
Yes. Temperature changes density and especially viscosity, which can substantially change Reynolds number.
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