Features of Turbulence

Energy Spectrum of Turbulence

Turbulence has a wide range of length (time) scales. A typical energy spectrum (Fourier decomposition of energy) is shown in the figure. Here  is the energy spectrum and  is wave number (inverse wavelength ( )). Fluctuation energy is produced at the large eddies (with low wave numbers). Vortex stretching mechanism then generates smaller and smaller eddies and energy flows down the spectrum to high wave number region. The energy is mainly dissipated into heat at the smallest eddies (of the order of the Kolmogorov scales).

The dissipation rate, , is roughly equal to the fluctuation energy production rate. Suppose the large-scale velocity fluctuation of turbulence is  and the corresponding length scale is . Then the rate of production (or dissipation) of fluctuation energy is given by

(1)

Equation (1) implies that large eddies lose a significant fraction of their energy in a time period of . Note that the direct viscous dissipation rate is

(2)

and the ratio

(3)

where

(4)

is a characteristic Reynolds number.