Reynolds Number

The Reynolds number is defined as \(\text{Re} = \frac{\rho u L}{\mu} = \frac{u L}{\nu}\), where \(\rho\) is the fluid density, \(u\) is a characteristic velocity, \(L\) is a characteristic linear dimension, \(\mu\) is the dynamic viscosity, and \(\nu\) is the kinematic viscosity. Inertial forces are related to the momentum of the fluid, which tend to cause fluid motion to persist, while viscous forces are frictional forces that tend to resist this motion and smooth out disturbances. When viscous forces dominate (low \(\text{Re}\)), any perturbations in the flow are damped out, resulting in a smooth, layered laminar flow. Conversely, when inertial forces dominate (high \(\text{Re}\)), small disturbances can grow and evolve into chaotic eddies and vortices, leading to turbulence.

The transition from laminar to turbulent flow is not abrupt but typically occurs over a range of Reynolds numbers. For flow in a pipe, this transition is generally observed around \(\text{Re} approx 2300-4000\). This transition is of immense practical importance; for example, turbulent flow in a pipe causes significantly higher frictional losses and requires more pumping power than laminar flow.

One of the most powerful applications of the Reynolds number is in the principle of dynamic similitude. If two geometrically similar flow situations have the same Reynolds number (and other relevant dimensionless numbers), their flow patterns will be dynamically similar. This allows engineers to test a small-scale model of an airplane in a wind tunnel and, by matching the Reynolds number, obtain results that accurately predict the aerodynamic forces on the full-scale aircraft.