Dynamic similarity (Reynolds and Womersley numbers)

In fluid mechanics, dynamic similarity is the phenomenon that when there are two geometrically similar vessels (same shape, different sizes) with the same boundary conditions (e.g., no-slip, center-line velocity) and the same Reynolds and Womersley numbers, then the fluid flows will be identical. This can be seen from inspection of the underlying Navier-Stokes equation, with geometrically similar bodies, equal Reynolds and Womersley Numbers the functions of velocity (u’,v’,w’) and pressure (P’) for any variation of flow.
Derivation.
The Reynolds number and the Womersley number are the only two physical parameters necessary to solve an incompressible fluid flow problem. The Reynolds number is given by:
The terms of the equation itself represent the following:
When the Reynolds number is large, it shows that the flow is dominated by convective inertial effects; When the Reynolds Number is small, it shows that the flow is dominated by shear effects.
The Womersley number is given by:
which is simply the square-root of the Stokes Number; the terms of the equation itself represent the following:
When the Womersley number is large (around 10 or greater), it shows that the flow is dominated by oscillatory inertial forces and that the velocity profile is flat. When the Womersley parameter is low, viscous forces tend to
dominate the flow, velocity profiles are parabolic in shape, and the center-line velocity oscillates in phase with the driving pressure gradient.
Starting with Navier–Stokes equation for Cartesian flow: