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The Turbulent Energy Spectrum

The time-averaged turbulent energy per unit mass, tex2html_wrap_inline2307 , 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 tex2html_wrap_inline2309 , 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 tex2html_wrap_inline2311 (which is inversely proportional to eddy sizegif) defined as:

  equation462

where tex2html_wrap_inline2315 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 tex2html_wrap_inline2317 , the total turbulent energy is:gif

  equation470

  
Figure 7.2: A log-log plot of the dimensionless turbulent energy spectrum tex2html_wrap_inline2321 as a function of the dimensionless wave number tex2html_wrap_inline2207 for a pipe flow at a Reynolds number of tex2html_wrap_inline2209 .

Figure 7.2 shows a sketch of the turbulent energy spectrum tex2html_wrap_inline2317 for a pipe flow at a Reynolds number of tex2html_wrap_inline2209 . The peak value of tex2html_wrap_inline2317 occurs at tex2html_wrap_inline2333 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 tex2html_wrap_inline2317 varies as tex2html_wrap_inline2337 , smaller eddies (greater wave number) contributing less to the total turbulent energy. At wave numbers greater than about tex2html_wrap_inline2339 , there is negligible energy because viscous dissipation causes the rapid decay of such small eddies.gif

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 tex2html_wrap_inline2347 for incompressible flow, equation 8.61 gif . When averaged over time, the rate of loss of turbulent energy, denoted by tex2html_wrap_inline2349 , can be expressed in the form:

  equation493

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 tex2html_wrap_inline2351 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 tex2html_wrap_inline2353 that is related to tex2html_wrap_inline2349 and the flow Reynolds number by:gif

  equation518

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 tex2html_wrap_inline2319 and tex2html_wrap_inline2349 , to characterize the effects of turbulence and be content to obtain approximate solutions to the mean flow field.


next up previous contents
Next: Turbulent Reynolds Stress Up: Characteristics of Turbulent Flow Previous: Eddy Description of Turbulent

Kavita Thomas
Fri Apr 5 16:29:53 EST 1996