How to estimate the Kolmogorov length scale

Question

My understanding of Kolmogorov scales doesn't really go beyond this poem:

Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity. - Lewis Fry Richardson

Th smallest whirl according to Wikipedia would be that big:

$\eta = (\frac{\nu^3}{\varepsilon})^\frac{1}{4}$

... with $\nu$ beeing kinematic viscosity and $\epsilon$ the rate of energy disspiation.

Since I find no straightforward way to calculate $\epsilon$, I'm completely at loss at what orders of magnitude to expect. Since I imagine this to be an important factor in some technical or biological processes, I assume that someone measured or calculated these microscales for real life flow regimes. Can anyone point me to these numbers?

I'm mostly interested in non-compressible fluids, but will take anything I get. Processes where I believe the microscales to be relevant are communities of synthropic bacteria (different species needing each others metabolism and thus close neighborhood) or dispersing something in a mixture.

Answer 1

The size of the Kolmogorv scale is not universal, it is dependent on the flow phenomena you are looking at. I don't know the details for compressible flows, so I will give you some hints on incompressible flows.

From the quotes poem, you can anticipate that everything that is dissipated at the smallest scales, has to be present at larger scale first. Therefor, as a very crude estimate, for a system of length $L$ and size $U$ (and dimensional grounds, on this scale viscosity does not play a role!), one could argue that

$$\varepsilon=\frac{U^3}{L}$$

For the crude estimate, one could use this $\varepsilon$ to estimate the Kolmogorov length scale.

To put in numbers, suppose you ($L=1m$) are running ($U=3m/s$) (in air $\nu=1.5\times10^{-5} m^2/s$), then, $\eta=100\mu m$. Which sounds at least reasonable.