Distorted vs undistorted model; Pi theorem

Distorted vs undistorted models

An undistorted model is geometrically similar to the prototype: all lengths (horizontal and vertical) have the same scale ratio. A distorted model has different scale ratios in different directions, usually a larger horizontal reduction than vertical.

Buckingham's $\pi$ theorem: working principle

Statement: If a physical phenomenon involves $n$ variables with $m$ fundamental dimensions (M, L, T), the variables can be arranged into $(n - m)$ independent dimensionless groups ($\pi$ terms), and the relation becomes

$$\phi(\pi_1, \pi_2, \dots, \pi_{n-m}) = 0$$

Procedure

1. List all variables on which the phenomenon depends, and count $n$.
2. Write the dimensions of each variable (M, L, T) and find $m$.
3. Choose $m$ repeating variables (a geometric one, a flow one and a fluid property, e.g. $D, V, \rho$) which together contain all the dimensions and are dimensionally independent.
4. Form $(n - m)$ $\pi$ terms, each containing the repeating variables raised to unknown powers and one non-repeating variable.
5. Find the powers by equating the exponents of M, L and T to zero.
6. Write $\pi_1 = f(\pi_2, \pi_3, \dots)$. Replace groups by standard ones (such as the Reynolds or Froude number) if needed.

The exact form of the function $f$ is then found by experiments on models, with the $\pi$ terms kept equal in model and prototype.