Reynolds number

A measure of the non-linearity of Navier-Stokes equations is given by the Reynolds number

$$Re = \frac{UL}{\nu}$$ (1.7)

where $L$ and $U$ are respectively the typical length scale and velocity of the fluid, e.g. in a pipe flow $L$ is the diameter of the pipe and $U$ the mean velocity. It was introduced by Osborne Reynolds, who showed that a transition between laminar and turbulent flow occurs when the $Re$ number reaches a critical value. Different geometries of the flow may change the critical $Re$ number, but the transition is universally controlled by this adimensional parameter. The Reynolds number plays a fundamental role in turbulence, since it gives a dimensional estimate of the relative weight between the inertial term ${\bf u} \cdot \nabla {\bf u}$ and the viscous term $\nu \Delta {\bf u}$:

$${[{\mbox{\boldmath$u$}} \cdot \nabla {\mbox{\boldmath$u$}}] \over [\nu \Delta {\mbox{\boldmath$u$}}]} \sim \frac{UL}{\nu}$$ (1.8)

Because of its definition, the limit $Re \to \infty$ in which fully developed turbulence is achieved, can be rephrased as the zero-viscosity limit $\nu \to 0$.