Kolmogorov Microscales

Kolmogorov microscales are the smallest scales in turbulent flow. At the Kolmogorov scale, viscosity dominates and the turbulent kinetic energy is dissipated into heat. They are defined by

where is the average rate of dissipation of turbulence kinetic energy per unit mass, and is the kinematic viscosity of the fluid.

In his 1941 theory, A. N. Kolmogorov introduced the idea that the smallest scales of turbulence are universal (similar for every turbulent flow) and that they depend only on and . The definitions of the Kolmogorov microscales can be obtained using this idea and dimensional analysis. It therefore is not a theory derived from first principles. Since the dimension of kinematic viscosity is length2/time, and the dimension of the energy dissipation rate per unit mass is length2/time3, the only combination that has the dimension of time is which is the Kolmorogov time scale. Similarly, the Kolmogorov length scale is the only combination of and that has dimension of length.

The Kolmogorov 1941 theory is a mean field theory since it assumes that the relevant dynamical parameter is the mean energy dissipation rate. In fluid turbulence, the energy dissipation rate fluctuates in space and time, so it is possible to think of the microscales as quantities that also vary in space and time. However, standard practice is to use mean field values since they represent the typical values of the smallest scales in a given flow.