Engineering & Technology · Fluid Mechanics, Flow Systems & Machines

Dimensional analysis, similarity and scaling

A small model can look exactly like a full-size object and still produce a different flow. Surface forces, inertia, gravity and compressibility do not all change in the same way when length changes. Scaling requires a physical argument as well as a drawing.

Review force vectors, energy and power, control volumes and steady-flow thermodynamics, SI units and measurement uncertainty. Be comfortable with ratios, areas, square roots and reading graphs. Symbols are defined where used: ρ is density, μ dynamic viscosity, Q volume flow and ṁ mass flow. For the constructed liquid cases, use g = 9.81 m/s² where stated. Prior physics study of fluids is helpful but not required.

Back to the chapter

After reading, you can

Explain what a scaled flow model must preserve and why matching one dimensionless number can conflict with another.

Remove units to expose comparisons

Dimensional analysis groups variables into combinations without units. For example, drag D divided by ½ρU²A gives a dimensionless drag coefficient CD. Since ρU²A has units kg/m³ × m²/s² × m² = N, the ratio is dimensionless. This rearrangement organizes evidence; it does not determine the coefficient or prove which variables can be ignored.

A characteristic length L must be defined consistently, such as pipe diameter or wing chord. Reynolds number Re = ρUL/μ compares inertial and viscous effects. Froude number Fr = U/√(gL) compares velocity with a gravity-based speed and is especially useful for free-surface flows. Its square gives an inertia-to-gravity force scale. Mach number M = U/a compares speed with local sound speed a and helps assess gas compressibility.

Similarity has several requirements

Geometric similarity preserves corresponding shape and length ratios, including relevant roughness and clearances. Kinematic similarity preserves corresponding velocity patterns and time scales. Dynamic similarity preserves the relative importance of the governing effects, expressed through the relevant independent dimensionless groups and boundary conditions.

Which groups matter depends on the question. A wave-making model needs gravity effects represented; a viscous creeping-flow model has different priorities. Surface tension, imposed unsteadiness or heat transfer can introduce further groups. Matching Re does not guarantee similarity if the model has different inlet turbulence, wall conditions or compressibility. Dimensional consistency is a necessary check, not a validation result.

A compromise needs a stated scope

If all important groups cannot be matched, a test may still answer a narrower question. The engineer identifies the dominant effects, records the mismatches, and checks sensitivity to them using additional evidence. A model that reproduces a free-surface pattern might not reproduce wall friction accurately. Its results should not silently become a complete full-scale performance prediction.

The plotted speed requirements assume the same fluid properties and gravity at every scale. They show why simply choosing a convenient model speed can preserve one effect while distorting another. Changing fluid, pressure or temperature can change the available scaling choices, but also changes material properties and test conditions that must be characterized.

Horizontal axis: model-to-prototype length ratio λ, 0.25 to 1. Vertical axis: required model speed, 0 to 9 m/s. Solid blue Reynolds matching follows 2/λ and rises to 8 m/s at quarter scale. Dashed orange Froude matching follows 2√λ and falls to 1 m/s at quarter scale. Both meet at λ = 1 and speed 2 m/s. The two requirements conflict at smaller scales with unchanged fluid properties.

Constructed quarter-scale model with competing requirements

Consider a prototype with Lp = 1 m, Up = 2 m/s, ρ = 1,000 kg/m³ and μ = 0.001 Pa·s. Its Re is 2,000,000. A geometrically similar model has Lm = 0.25 m in the same fluid and gravity. Matching Reynolds number requires UmLm = UpLp, so Um = 8 m/s.

Matching Froude number instead requires Um/Up = √(Lm/Lp) = 0.5, so Um = 1 m/s. At this speed, model Re = 250,000, only one-eighth of prototype Re. Prototype and model Fr both equal approximately 0.639. The two speed demands conflict. Choosing Froude matching for a gravity-dominated question requires examining whether the Reynolds mismatch changes the measured behaviour; large Re alone is not a universal exemption.

Try the reasoning

Why is a coefficient without units still condition-dependent?
Show the answer and reasoning
Removing units does not remove physics. CD can vary with Re, Mach number, shape, roughness and orientation, among other conditions. A dimensionless value needs a defined scope.
At half scale in the same fluid, what speed matches Re for the 2 m/s prototype?
Show the answer and reasoning
4 m/s, because speed must double when characteristic length halves to keep ρUL/μ unchanged. This does not automatically match gravity or compressibility effects.
Would matching the model’s visible wave height alone establish dynamic similarity?
Show the answer and reasoning
No. One visible agreement can hide mismatched forces, velocities or boundaries. It supports only the measured comparison unless further evidence establishes the intended similarity.

What to carry forward

Match the physics that controls the intended result, and disclose the effects a scaled model cannot preserve.

References

Teaching resources and research supporting this lesson’s explanations and examples.