Kolmogorov turbulence spectrum

Note that the Kolmogorov power spectrum is unphysical at low frequencies— the variance is infinite at k = 0. In fact the turbulence is only homogeneous within a finite range—the inertial subrange. The modified von Karman spectral model includes effects of finite inner and outer scales. 

Here is the root-mean-square (rms) relative velocity between two points in the fluid separated by a distance d. For very large Reynolds numbers of the main stream (much larger than the Re value required for assumption of universal equilibrium), Kolmogorov s theory proposes that the turbulence spectrum be divided into two subranges. The inertial subrange is that part of the spectrum in which viscous dissipation is unimportant and... 

Davies (Turbulence Phenomena, Academic, New York, 1972) presents a good discussion of the spectrum of eddy lengths for well-developed isotropic turbulence. The smallest eddies, usually called Kolmogorov eddies (Kolmogorov, Compt. Rend. Acad. Sci. URSS, 30, 301 32, 16 [1941]), have a characteristic velocity fluctuation given by... 

The h rpotheses of Kolmogorov allow a number of additional deductions to be formulated on the statistical characteristics of the small-scale components of turbulence. The most important of them is the two-third-law deduced by Kolmogorov [84]. This law states that the mean square of the difference between the velocities at two points of a turbulent flow, being a distance x apart, equals C(ex) / when x lies in the inertial subrange. (7 is a universal model constant. Another form of this assertion (apparently first put forward by Obukhov [116] [117]) is the five-third law. This law states that the spectral density of the kinetic energy of turbulence over the spectrum of wave numbers, k, has the form Cke / k / in the inertial subrange. Cj, is a new model constant (see e.g., [8], sect. 6.4). 

Kolmogorov s theory establishes that the energy spectrum has a universal form. In other words, the energy spectrum varies with the wave number in identical fashion in all turbulent flows, provided that the (k and it scales are known. 

Figure 11.4. Turbulent energy spectrum according to Kolmogorov s theory...

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