Reynolds number#

Named after: Osborne Reynolds (1842-1912).

$$\text{Re} \stackrel{\text{def}}{=} \frac{\rho U L}{\mu} \sim \frac{\text{inertia}}{\text{viscosity}}$$

Measures inertial transport relative to viscous resistance. It is the primary indicator of laminar, transitional, and turbulent flow behavior.

Name	Symbol	SI units	Dimension
mass density	$\rho$	$\mathrm{kg}\,\mathrm{m}^{-3}$	$\text L^{-3}\,\text M$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$
characteristic length	$L$	$\mathrm{m}$	$\text L$
dynamic viscosity	$\mu$	$\mathrm{Pa}\,\mathrm{s}$	$\text L^{-1}\,\text M\,\text T^{-1}$

Pipe flow

Range	Regime	Description
0 – 2300	laminar	Smooth, ordered flow. Viscous forces dominate. Velocity profile is parabolic in pipes.
2300 – 4000	transitional
4000 – ∞	turbulent	Chaotic, diffusive flow. Inertial forces dominate. Enhanced mixing and heat transfer.

Name	Symbol	SI units	Dimension
mass density	\(\rho\)	\(\mathrm{kg}\,\mathrm{m}^{-3}\)	\(\text L^{-3}\,\text M\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)
dynamic viscosity	\(\mu\)	\(\mathrm{Pa}\,\mathrm{s}\)	\(\text L^{-1}\,\text M\,\text T^{-1}\)