Reynolds number

Remarks

It is a dimensionless number: $[U L] = \mathsf{L}^2 \mathsf{T}^{-1} = [\nu]$. Example: water with $\nu \approx 10^{-6}\ \mathrm{m^2/s}$ flowing at $U = 1\ \mathrm{m/s}$ past an object of size $L = 1\ \mathrm{cm}$ has $\mathrm{Re} \approx 10^{4}$. Non-example: the ratio $U / \nu$ is not a Reynolds number; it has dimension $\mathsf{L}^{-1}$ because the length scale is missing. In the variables $x' = x/L$, $t' = t U / L$, $\mathbf{u}' = \mathbf{u}/U$, the incompressible Navier–Stokes equations contain the viscous term with the factor $1/\mathrm{Re}$ and no other parameter. The value of $\mathrm{Re}$ depends on the choice of $U$ and $L$, so a value is meaningful only together with this choice. In pipe flow, for example, one uses the diameter and the mean speed.