## Kolmogorov Microscales

In the view of Kolmogorov (**Andrey Nikolaevich Kolmogorov** was a Russian mathematician who made significant contributions to the mathematics of probability theory and turbulence), turbulent motions involve a **wide range of scales**. From a **macroscale** at which the energy is supplied, to a **microscale** at which energy is dissipated by viscosity.

For example, consider a cumulus cloud. The **macroscale** of the cloud can be of the order of** kilometers** and may grow or persist over long periods of time. Within the cloud, **eddies** may occur over scales of the order of **millimeters**. For smaller flows such as in pipes, the microscales may be much smaller. Most of the kinetic energy of the turbulent flow is contained in the macroscale structures. The energy “cascades” from these macroscale structures to microscale structures by an inertial mechanism. This process is known as the **turbulent energy cascade**.

The **smallest scales** in turbulent flow are known as the **Kolmogorov microscales**. These are small enough that molecular diffusion becomes important and **viscous dissipation of energy** takes place and the turbulent kinetic energy is dissipated into heat.

The smallest scales in turbulent flow, i.e. the **Kolmogorov microscales** are:

where ε is the average rate of dissipation rate of turbulence kinetic energy per unit mass and has dimensions (m^{2}/s^{3}). ν is the kinematic viscosity of the fluid and has dimensions (m^{2}/s).

The size of the smallest eddy in the flow is determined by viscosity. The** Kolmogorov length** scale decreases as viscosity decreases. For very high **Reynolds number** flows, the viscous forces are smaller with respect to inertial forces. Smaller scale motions are then necessarily generated until the effects of viscosity become important and energy is dissipated. The ratio of largest to smallest length scales in the turbulent flow are proportional to the Reynolds number (raises with the **three-quarters power**).

This causes direct numerical simulations of turbulent flow to be practically impossible. For example, consider a flow with a **Reynolds number** of** 10 ^{6}**. In this case the ratio L/l is proportional to 10

^{18/4}. Since we have to analyze three-dimensional problem, we need to compute a grid that consisted of at least

**10**. This far exceeds the capacity and possibilities of existing computers.

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