The time-averaged turbulent energy per unit mass, , is the combination of the kinetic energies of many eddies of different sizes. The allocation of this kinetic energy to motions of different length scales is called the turbulent energy spectrum. A knowledge of this spectrum can tell us something about the details of the turbulent motion.
An energy spectrum with which we are undoubtedly familiar is that of sunlight. If sunlight is decomposed into its constituents of different wavelengths by reflection from a diffraction grating, the energy flux in each wavelength, or frequency band, can be measured. When this is done, it is found that the peak energy flux occurs at about a wavelength of , with much lower values encountered at longer (infrared) and shorter (ultraviolet) wavelengths. This distribution reflects the thermal equilibrium in the sun's photosphere, the origin of sunlight.
The distribution of turbulent kinetic energy among motions of different length scales is commonly given as a function of the wave number (which is inversely proportional to eddy size) defined as:
where is the angular frequency of the components of the turbulent velocity history illustrated in figure 7.1. Denoting this turbulent energy per unit wave number by , the total turbulent energy is:
Figure 7.2: A log-log plot of the dimensionless turbulent energy spectrum as a function of the dimensionless wave number for a pipe flow at a Reynolds number of .
Figure 7.2 shows a sketch of the turbulent energy spectrum for a pipe flow at a Reynolds number of . The peak value of occurs at at the low wave number end of the spectrum, where most of the turbulent energy is associated with the biggest eddies (lowest wave number). Over most of the spectral range varies as , smaller eddies (greater wave number) contributing less to the total turbulent energy. At wave numbers greater than about , there is negligible energy because viscous dissipation causes the rapid decay of such small eddies.
The amount of turbulent energy in a flow is limited by the loss of energy due to viscous dissipation. We can calculate the rate at which turbulent energy is lost from a knowledge of the velocity derivatives in the form of the dissipation function for incompressible flow, equation 8.61 . When averaged over time, the rate of loss of turbulent energy, denoted by , can be expressed in the form:
Notice that the principal contribution to the integral on the right side of 7.10 comes at the high wave number end of the spectrum of figure 7.2 where reaches its maximum. Thus the smallest eddies contribute the most to the energy dissipation while the largest eddies contribute most to the turbulent energy.
Kolmogoroff showed that the smallest eddy has a wave number that is related to and the flow Reynolds number by:
The smallest eddies are therefore several orders of magnitude smaller than the largest, energy containing eddies since the Reynolds number of a turbulent flow is necessarily large. This fundamental physical property of turbulent flow makes it extremely difficult to describe completely a turbulent flow, even using the largest and fastest computers available today. Instead, we must compromise by using much less information, such as and , to characterize the effects of turbulence and be content to obtain approximate solutions to the mean flow field.