Guide
Non-dimensional numbers describe a flow independently of units. They tell you which physical effect dominates, which model is appropriate, and whether a laboratory model will behave like the full-scale system. The calculator above covers nine of the most useful ones, with formulas, definitions and regime thresholds. This page is a quick reference.
If you write the governing equations in terms of scaled variables, only a few dimensionless groups remain. Two systems that share the same values of these groups are dynamically similar, whatever their size or fluid, which is the idea behind the Buckingham Π theorem. That is why you can test a scale model in a wind tunnel or water channel and apply the result to the real thing, and why a CFD case is often more informative when its results are reported as dimensionless quantities.
| Number | Definition | Meaning | Rules of thumb |
|---|---|---|---|
| Reynolds Re | ρUL / μ | Inertial to viscous forces | Pipe: laminar below about 2,300; turbulent above about 2,900. Flat plate: transition near 5×10⁵ |
| Mach Ma | U / a | Flow speed to speed of sound (compressibility) | Below 0.3 essentially incompressible; 0.8–1.2 transonic; above 1 supersonic; above 5 hypersonic |
| Prandtl Pr | cpμ / k = ν / α | Momentum to thermal diffusivity | Air ≈ 0.71; water ≈ 7 at 20 °C; liquid metals ≪ 1; oils ≫ 100 |
| Schmidt Sc | ν / D | Momentum to mass diffusivity | Gases ≈ 0.2–2; liquids around 10²–10³ |
| Péclet Pe | UL / α (thermal) or UL / D (mass) = Re·Pr or Re·Sc | Advection to diffusion | Pe ≫ 1: advection dominated; Pe ≪ 1: diffusion dominated |
| Strouhal St | fL / U | Unsteady oscillation to convective time | Vortex shedding behind a cylinder: St ≈ 0.2 over a wide range of Re |
| Froude Fr | U / √(gL) | Inertia to gravity | Free-surface flows: Fr < 1 subcritical, Fr > 1 supercritical |
| Weber We | ρU²L / σ | Inertia to surface tension | Droplet secondary breakup begins near We ≈ 12; higher We means more violent breakup |
| Knudsen Kn | λ / L | Molecular mean free path to length scale | Kn < 0.01 continuum; 0.01–0.1 slip; 0.1–10 transitional; above 10 free molecular |
Symbols: ρ density, U velocity, L characteristic length, μ dynamic viscosity, ν kinematic viscosity, a speed of sound, cp specific heat, k thermal conductivity, α thermal diffusivity, D mass diffusivity, f frequency, g gravitational acceleration, σ surface tension, λ molecular mean free path.
| Example | Calculation | Result |
|---|---|---|
| Airflow at 100 m/s in air at 20 °C (a ≈ 343 m/s) | Ma = 100 / 343 | Ma ≈ 0.29: at the edge of the incompressible range |
| Vortex shedding from a 20 mm cylinder in a 5 m/s flow | f = St U / L = 0.2 × 5 / 0.02 | ≈ 50 Hz |
| Water at 20 °C: ν = 1.0×10⁻⁶ m²/s, α = 1.43×10⁻⁷ m²/s | Pr = ν / α | ≈ 7: the thermal boundary layer is thinner than the velocity boundary layer |
| Diesel droplet, ρ = 830 kg/m³, U = 100 m/s, d = 50 µm, σ = 0.028 N/m | We = ρU²d / σ | ≈ 1.5×10⁴: far above the breakup threshold |
To scale an experiment, list the dimensionless groups relevant to your physics (for example Re for viscous flow, Fr for free surfaces, We for droplets) and try to match them between model and prototype. It is usually impossible to match all of them at once. In practice you match the dominant group and confirm that the others lie in the same regime. The calculator’s Theory tab explains each group, and the Calculator tab lets you compute values from your own inputs.
Use the length that governs the physics and state it clearly: hydraulic diameter for ducts, plate length for boundary layers, cylinder diameter for bluff bodies, droplet diameter for We and Oh. Published thresholds assume specific definitions.
The thermal Péclet number equals Re × Pr, and the mass-transfer Péclet number equals Re × Sc.
The current version covers the nine numbers listed above. If you would like others added, get in touch through the community forum.
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