Reynolds number

The Reynolds number $\mathrm{R}e$ is a dimensionless quantity that is used to help predict similar flow patterns in different fluid flow situations. The concept was introduced by George Gabriel Stokes in 1851, but the Reynolds number is named after Osborne Reynolds (1842–1912), who popularized its use in the 1880s.

The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.

$Re\mathrm{=}\frac{{\rho }vD}{{\mu }}$

(1)

It should be noted that $D$ is the inner diameter of the pipe, and $v$ is the average velocity of the fluid. The Reynolds number is used to characterize different flow regimes within a similar fluid, such as laminar or turbulent flow. Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant. Laminar flow is characterized by smooth, constant fluid motion. Turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

By defining the following non-dimensional variables $\begin{array}{rl}{\mathbf{v}}^{\prime }& =\frac{\mathbf{v}}{v}\\ p^{\prime }& =\frac{p}{\rho v^2}\\ {\mathbf{g}}^{\prime }& =g\frac{D}{v^2}\\ {\mathrm{\nabla }}^{\prime }& =D\mathrm{\nabla }\\ \frac{\mathbf{D}}{\mathbf{D}{t}^{\prime }}& =\frac{D}{v}\left(\frac{\mathbf{D}}{\mathbf{D}t}\right)\end{array}$ and substuting into the Navier-stokes equation, we get $${{{\bf D}{\bf{v'}}} \over {{\bf D}t'}} = - \nabla 'p' + {\mu \over {\rho Dv}}{\nabla '^2}{\bf{v'}} + {\bf{g'}}$$ or $${Template:\bf Dv'\over{{\bf D}t'}}-{{1}\over{Re}}{\nabla'}^{2}{\bf v'}=-\nabla'p'+{\bf g'}$$ At low Reynolds numbers, \({Template:\bf Dv'\over{{\bf D}t'}}\ll{{1}\over{Re}}{\nabla'}^{2}{\bf v'}\nonumber\) and $$-{{1}\over{Re}}{\nabla'}^{2}{\bf v'}=-\nabla'p'+{\bf g'}$$ which reduces to Eq. \eqref{eq.laminar} . At high Reynolds numbers \({Template:\bf Dv'\over{{\bf D}t'}}\gg{{1}\over{Re}}{\nabla'}^{2}{\bf v'}\nonumber\)and $${Template:\bf Dv'\over{{\bf D}t'}}=-\nabla'p'+{\bf g'}$$ This limit permits solutions in which backward flow is possible, which gives rise to the instabilities that lead to turbulent flow.