Dimensional analysis

Dimensional analysis uses the fact that every valid physical equation must be dimensionally consistent to reduce the number of variables in a problem. Instead of studying how pipe friction depends separately on velocity, diameter, roughness, density and viscosity, we find a few dimensionless groups, such as Reynolds number and relative roughness, and study how the dimensionless head loss depends on those. Fewer variables mean fewer experiments and results that apply at any scale.

Buckingham's Pi theorem states that a problem with n variables and m fundamental dimensions (mass, length, time) can be described by n − m independent dimensionless groups. Rayleigh's method is a simpler approach for problems with few variables. The common groups in hydraulics are the Reynolds number (inertia to viscous forces), the Froude number (inertia to gravity), the Weber number (inertia to surface tension), the Euler number (pressure to inertia) and the Mach number (inertia to elasticity).

This underpins hydraulic model studies. A scale model of a spillway, harbour or river reach behaves like the prototype if the governing dimensionless number is the same in both. Open channel and free-surface flows are dominated by gravity, so Froude similarity is used: velocities scale with the square root of the length scale and discharges with the length scale to the power 2.5. Exact similarity of all numbers at once is rarely possible, so scale effects are judged and models are kept large enough that viscous and surface tension effects are small.