Kolmogorov microscales

Remarks

They are the unique combinations of $\nu$, with units $\mathrm{m^2/s}$, and $\varepsilon$, with units $\mathrm{m^2/s^3}$, with units of length, time and velocity. They satisfy $\eta = u_\eta \tau_\eta$ and $u_\eta \eta / \nu = 1$, so the Reynolds number of the smallest eddies is $1$, the balance of inertia and viscosity. If $L$ is the size of the large eddies and $u$ their speed, with $\varepsilon \sim u^3 / L$, then $L / \eta \sim \mathrm{Re}^{3/4}$ with $\mathrm{Re} = u L / \nu$. So a full simulation in three dimensions needs of the order of $\mathrm{Re}^{9/4}$ grid points.

Example: in air with $\nu = 1.5 \times 10^{-5}\ \mathrm{m^2/s}$ and $\varepsilon = 1\ \mathrm{m^2/s^3}$, one gets $\eta \approx 0.24\ \mathrm{mm}$, $\tau_\eta \approx 3.9\ \mathrm{ms}$ and $u_\eta \approx 6.2\ \mathrm{cm/s}$; indeed $0.24\ \mathrm{mm} / 3.9\ \mathrm{ms} \approx 6.2\ \mathrm{cm/s}$. Non-example: the quantity $(\nu / \varepsilon)^{1/4}$ is not a microscale; it has units $\mathrm{s^{1/2}}$, neither a length nor a time.