Similitude, Scaling & Model Testing

Model tests of ships, aircraft, dams and buildings work only if the model is geometrically, kinematically and dynamically similar to the prototype. Learn scale ratios, Reynolds and Froude model laws, how to scale velocities, forces and power, what happens when similarity is incomplete, and practical tricks like changing the test fluid.

You cannot build a full-size ship to see how much power it needs. Instead you tow a 1:25 model in a tank. Dimensional analysis tells you which speeds to test and how to scale the measured drag up to the real ship.

The catch: usually you cannot match every dimensionless group at once. Engineers choose the dominant one (Froude for waves, Reynolds for viscous drag, Mach for compressibility) and correct for the rest.

An analogy

A scale model is like a translated sentence: the words (dimensions) change, but if the grammar (dimensionless groups) is preserved, the meaning (behaviour) carries over.

Three levels of similarity

All three are needed for results to transfer.

Froude scaling for free-surface flows

Waves, spillways and ship wave-making are gravity-dominated: match Fr = V/√(gL). Velocities scale as √λ, time as √λ, forces as λ³ and power as λ^3.5 (same fluid). A 1:25 model of a 10 m/s ship is towed at 2 m/s.

Reynolds scaling for enclosed and aerodynamic flows

Pipe flow, low-speed aerodynamics and viscous drag need equal Re. In the same fluid, a 1:5 model needs 5× the speed — often impractical or compressible. Solutions: pressurised wind tunnels (higher ρ), cryogenic tunnels (lower μ), or water tunnels (ν ≈ 1/15 of air).

Incomplete similarity

Matching both Fr and Re with a small model would need a fluid with ν_m = ν_p/λ^1.5 — impossible for ships. Froude's method splits resistance: friction is scaled with Re-based flat-plate correlations (ITTC-57 line), the residual (wave) part by Froude scaling. Understanding which groups dominate is the engineering judgment.

Formulas

Worked example: Ship model (Froude)

Given: Prototype 200 m, 10 m/s; λ = 25, same water

Find: Model length and speed.

Answer: 8 m at 2 m/s (Fr = 0.226 both)

Worked example: Car in a water tunnel (Reynolds)

Given: Car 4 m at 30 m/s in air (ν = 1.5×10⁻⁵); 1:5 model in water (ν = 1×10⁻⁶)

Find: Model water speed.

Answer: V_m = 10 m/s (air model would need 150 m/s!)

Worked example: Scaling a measured force

Given: Model wave force 10 N; λ = 25, same fluid

Find: Prototype force.

Answer: F_p ≈ 156 kN

Common mistakes

Applications

Key takeaways

Going further