Dimensionless Numbers: Definition and Examples
A dimensionless quantity is commonly formed by dividing quantities with the same dimensions or by combining variables so every dimensional exponent cancels. Its coherent SI unit is one, symbol 1, although the symbol is normally omitted from the reported value.
A basic ratio example
Strain compares a change in length ΔL with an original length L:
strain = ΔL ÷ L.
If a 2.000 m bar lengthens by 1.0 mm, first use the same unit for both values: 2.000 m = 2,000 mm. The strain is 1.0 ÷ 2,000 = 0.0005. It may also be expressed as 0.05%, but percent is a scaled form of the same ratio.
Reynolds number
Reynolds number compares inertial effects with viscous effects in fluid flow:
Re = ρvL ÷ μ, where ρ is density, v is speed, L is a characteristic length, and μ is dynamic viscosity.
When SI units are substituted, kg/m³ × m/s × m ÷ (kg/(m·s)) reduces to one. Reynolds number helps compare flows and anticipate whether viscous or inertial behavior dominates, but transition thresholds depend on geometry and conditions.
Mach number
Mach number is the ratio of an object or flow speed v to the local speed of sound a:
M = v ÷ a.
If an aircraft travels at 250 m/s where the local speed of sound is 340 m/s, M ≈ 0.74. Both inputs use m/s, so the units cancel. The speed of sound changes with the medium and thermodynamic state, so Mach number cannot be inferred from speed alone.
Other useful examples
- Refractive index: a ratio of two speeds.
- Coefficient of friction: friction force divided by normal force.
- Relative density: density compared with a reference density.
- Prandtl number: a ratio comparing momentum and thermal diffusivity.
- Probability: a number between zero and one, often shown as a percentage.
Why engineers use dimensionless groups
- Compare systems of different physical sizes.
- Design scale-model experiments with similar governing behavior.
- Reduce the number of independent variables in an analysis.
- Present correlations that work across consistent unit systems.
Common mistakes
- Entering inconsistent units before cancellation, such as millimetres for one length and metres for another.
- Assuming “dimensionless” means physically meaningless.
- Treating degrees or percent as independent dimensions rather than scaled conventions.
- Applying a threshold from one geometry or flow regime to a different problem.