Reynolds number is a dimensionless value that predicts how a fluid is likely to flow. It compares the tendency of a moving fluid to keep moving with the tendency of viscosity to smooth out motion. Engineers use it to decide whether flow in a pipe, channel, around a wing, or past a vehicle will be smooth, transitional, or turbulent.
This matters because flow type strongly affects drag, pressure loss, mixing, heat transfer, and pump power.
The standard form is Re = ρvL / μ = vL / ν, where ρ is density, v is characteristic speed, L is characteristic length, μ is dynamic viscosity, and ν is kinematic viscosity. Low Reynolds number means viscous forces dominate, so fluid layers slide smoothly in laminar flow. High Reynolds number means inertial forces dominate, so small disturbances grow into turbulent eddies.
In pipe flow, values below about 2300 are usually laminar, values from about 2300 to 4000 are transitional, and values above about 4000 are usually turbulent.
Key Facts
- Re = ρvL / μ = vL / ν
- Reynolds number has no units because it is a ratio of inertial effects to viscous effects.
- For circular pipe flow, laminar flow usually occurs when Re < 2300.
- For circular pipe flow, transitional flow usually occurs when 2300 < Re < 4000.
- For circular pipe flow, turbulent flow usually occurs when Re > 4000.
- Increasing speed v, length scale L, or density ρ increases Re, while increasing viscosity μ decreases Re.
Vocabulary
- Reynolds number
- A dimensionless number that compares inertial forces to viscous forces in a flowing fluid.
- Laminar flow
- A smooth flow pattern in which fluid moves in orderly layers with little mixing between layers.
- Turbulent flow
- An irregular flow pattern with swirling eddies, strong mixing, and rapidly changing velocity.
- Dynamic viscosity
- A measure of a fluid's resistance to shearing motion, represented by μ.
- Kinematic viscosity
- Dynamic viscosity divided by density, represented by ν, so ν = μ / ρ.
Common Mistakes to Avoid
- Using diameter only for every flow problem, which is wrong because L must be the characteristic length that matches the geometry, such as pipe diameter, channel hydraulic diameter, or object length.
- Forgetting that Reynolds number is dimensionless, which is wrong because the units cancel when ρvL / μ or vL / ν is formed correctly.
- Treating 2300 and 4000 as exact universal cutoffs, which is wrong because transition depends on surface roughness, disturbances, geometry, and inlet conditions.
- Assuming higher viscosity increases Reynolds number, which is wrong because viscosity is in the denominator, so more viscosity makes viscous forces more important and lowers Re.
Practice Questions
- 1 Water with density 1000 kg/m^3 and dynamic viscosity 0.001 Pa·s flows through a pipe of diameter 0.050 m at a speed of 0.20 m/s. Calculate Re and classify the flow as laminar, transitional, or turbulent.
- 2 Air with kinematic viscosity 1.5 × 10^-5 m^2/s flows past a model car of length 0.30 m at 12 m/s. Calculate Re using Re = vL / ν.
- 3 Two fluids flow through identical pipes at the same speed, but one fluid has a much larger dynamic viscosity. Explain how the Reynolds number and expected flow behavior change.