What are the length and time scales in turbulence?

Question

I haven't been able to understand what are does someone mean by length and time scales, while talking about turbulence. Can someone explain it?

Answer 1

Basically, the scale of a certain parameter is the order of magnitude of that parameter. Being able to determine the scales of a parameters in a complex system (like turbulence problems) is very useful.

For turbulence, the size of the largest eddies is given by the characteristic length scale you are working with, $L$, and the smallest eddy size is given by the so called, Kolmogorov length scale, $\eta$.

This scale goes like, $\eta = \left(\frac{\nu^3}{\epsilon}\right)^{1/4}$, where $\nu$ is the viscosity and $\epsilon$ is the dissipation rate per unit mass. It is interesting to note that $L/\eta= \text{Re}^{3/4}$.

Another commonly encountered length scale is known as Taylor micro-scale and provides a good estimate for the fluctuating strain rate field.

The times scale for the so called "large eddy turnover" is simply the time scale of the flow, $L/U$

The time scales for the small eddies can also be derived using the viscosity and dissipation, $t_\eta = \left(\frac{\nu}{\epsilon}\right)^{1/2}.$

Similarly, $t_L/t_\eta= \text{Re}^{1/2}.$

Good books I can suggest are the book S. B. Pope (Turbulent Flows) and T.H. Lumely (A first course in Turbulence).