What is the Reynolds number?
The Reynolds number compares the inertial forces in a flow with its viscous forces. At low Reynolds numbers viscosity dominates and the flow is smooth and layered (laminar); at high Reynolds numbers inertia dominates and the flow becomes turbulent.
Here ρ is the density, U the velocity, L the characteristic length, μ the dynamic viscosity and ν the kinematic viscosity. Because it is dimensionless, the same Reynolds number means the same flow behaviour at any scale, which is why scaled models and wind tunnel tests match the Reynolds number of the real product.
Choosing the characteristic length
The characteristic length, and the Reynolds number where the flow changes, depend on the geometry:
| Geometry | Characteristic length | Where the flow changes |
|---|---|---|
| Pipe | Inside diameter D | Where the flow changes: Laminar below 2,300, turbulent above about 4,000 |
| Non-circular duct | Hydraulic diameter Dₕ = 4A / P | Where the flow changes: As for pipes, using Dₕ |
| Flat plate | Distance x from the leading edge | Where the flow changes: Transition from about Reₓ = 5 × 10⁵ |
| Cylinder or sphere | Diameter D | Where the flow changes: Drag crisis around Re ≈ 3 × 10⁵ |
Hydraulic diameter for ducts
For a duct that is not round, use the hydraulic diameter, four times the flow area divided by the wetted perimeter. The calculator works it out for rectangular ducts and annuli:
Fluid properties used
For air, the density comes from the ideal gas law (R = 287.058 J/(kg·K)) and the viscosity from Sutherland's law, accurate for dry air from about −100 °C to 1,000 °C at moderate pressures. For liquid water at atmospheric pressure, the density comes from the Kell equation and the viscosity from the Vogel equation, between 0 °C and 100 °C. For any other fluid, enter its density and its dynamic or kinematic viscosity.
Why the Reynolds number matters in CFD
The Reynolds number decides whether a simulation needs a turbulence model, which model suits it, and how fine the mesh must be near the walls. Once you know it, our y+ calculator gives the first cell height for the boundary layer. For example, our Ahmed body CFD ran at Reynolds numbers of 1–2 million with the realizable k-ε model. For a complete study, see our CFD analysis services.
Frequently asked questions
How do I calculate the Reynolds number?
Multiply the density by the velocity and the characteristic length, then divide by the dynamic viscosity: Re = ρUL/μ, or Re = UL/ν with the kinematic viscosity. For water at 20 °C (ρ = 998 kg/m³, μ = 1.0 × 10⁻³ Pa·s) flowing at 1 m/s in a 50 mm pipe, Re ≈ 49,800, so the flow is turbulent.
At what Reynolds number is flow laminar?
In a pipe, flow is laminar below about 2,300 and turbulent above about 4,000. Over a flat plate, the boundary layer is usually laminar below Reₓ ≈ 5 × 10⁵. The exact values depend on surface roughness, inlet conditions and disturbances.
What is the critical Reynolds number?
The value at which the flow starts to change character: about 2,300 in pipes, about 5 × 10⁵ for a flat plate boundary layer, and about 3 × 10⁵ for the drag crisis on cylinders and spheres.
What is the difference between dynamic and kinematic viscosity?
Dynamic viscosity μ (Pa·s) measures a fluid's resistance to shearing; kinematic viscosity ν (m²/s) is the dynamic viscosity divided by the density, ν = μ/ρ. The Reynolds number can use either: Re = ρUL/μ = UL/ν.
Do I need a turbulence model for my CFD simulation?
If the Reynolds number puts your flow in the turbulent range, yes, for example k-ω SST or k-ε. Laminar flows can be solved without a turbulence model, and transitional flows may need a transition model such as γ-Reθ.