dimensional analysis Buckingham Pi
With one small model ship you can predict the resistance of a huge real ship. The secret is a dimensionless number whose units all cancel, and drag the speed to see how the units of the Reynolds number disappear. A number whose units vanished ties the model to the real thing.
Look at the Reynolds number Re = ρvLμ. The units of the numerator ρvL are kg·m⁻¹·s⁻¹ and the units of the denominator μ are the same kg·m⁻¹·s⁻¹. So when you divide, the units cancel whole and only a pure number remains, and when you drag the speed v the value of Re changes but it is always a dimensionless number with no unit. This number whose units vanished holds the nature of the flow. The top and the bottom share the same units, so only the quotient remains.
The dimensionless numbers that govern a flow are all ratios of competing forces. Compare the three with a button. The Reynolds number Re = ρvLμ is inertia against viscosity, the Froude number Fr = v√(gL) is inertia against gravity, and the Mach number Ma = vc is flow speed against the speed of sound, compressibility. Which force wins sets the character of the flow. Pipes, wings, and swimming are the first. Ships and channel waves are the second. Jets and supersonic flow are the third.
Feel the Froude number as a ship. As the ship goes faster the wave the bow makes grows long and fierce, and when Fr = v√(gL) nears 1 the ship struggles to climb over the bow wave it made itself. That is the hull's limiting speed. Drag the speed and you see the wave and Fr grow together, so a model-ship experiment matches Fr to the real ship. The same number makes the same wave.
When the dimensionless numbers are the same, the two flows are alike, and that is dynamic similarity. To match the Re of the real thing and the model, make the flow faster by as much as you made the model smaller, keeping v·L constant, and when you drag the model size the wind speed rises as it shrinks and the two Re stay equal. So a small model in a wind tunnel reproduces the flow of a real airplane. Faster by as much as it shrank, and it is the same flow.
The last reward. Countless drag measurements of every size and speed, drawn in dimensionless form, collapse onto one curve CD(Re), and when you drag Re the point moves along this universal curve. At low Re viscosity dominates and CD is large, in the middle it is flat, and near the critical Re the boundary layer becomes turbulent and a drag crisis appears, a sharp drop. The Moody chart and wind-tunnel data are both this dimensionless curve, and dimensional analysis ties pressure and flow, viscosity, the boundary layer, and drag into one language and closes the whole of fluid mechanics.