Kolmogorov turbulence theory

The smallest size for turbulent eddies is given by the Kolmogorov microscale A. Energy loss below this size only occurs via viscous dissipation. Here also, several scales have been introduced in the framework of turbulence theory, depending whether velocity or concentration fluctuations are considered, namely Ak> Ag, and Ac (see Table I). In liquids, Ak is typically between 10 and 100 ym. The Kolmogorov microscale Ak is frequently used in the in-... 

If the mass or heat transfer in the S/L system is considered from the standpoint of Kolmogorov s theory of isotropic turbulence, it can be expected that both transfer coefficients only depend upon the stirrer power dissipated in the system and the kinematic viscosity of the fluid. In the state of suspension the mass transfer must then be represented in the pi space ... 

Under the conditions under which Kolmogorov s theory of isotropic turbulence [289] applies, the state of turbulence depends upon the local contribution of energy dissipation per unit mass of the liquid (e = P/p V) ... 

The description of turbulent mixing of passive scalars is based on an extension of the turbulence theory developed for the momentum transfer processes, described in sect 1.2.7. In particular, the energy cascade idea introduced by Richardson [76] and the Kolmogorov hypotheses [47] are adopted. 

Although the primary energy input in most multiphase reactors results in anisotropic turbulence, the ultimate dissipation of the input energy by viscous dissipation through the smallest eddies yields isotropic turbulence. This matter will be discussed in some detail under Kolmogorov s theory. 

Approach Based on Kolmogorov s Theory The first attempt at putting forth a credible theory of turbulence is attributed to Kohnogorov (1941). A simple... 

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... 

In conclusion, Kolmogorov s theory is a realistic description of decay of turbulence. However, its application in a sophisticated form of dimensional analysis (Middleman 1965) is stretching the theory beyond its aim. Such attempts and the resulting correlations cannot lead to a unique scheme for predicting transport properties. 

Correlation based on Kolmogorov s theory of isotropic turbulence. Drawbacks discussed in Section 6.3.2. Does not include term... 

Kawase Y, Moo-Young M. (1990) Mathematical models for design of bioreactors applications of Kolmogorov s theory of isotropic turbulence. Chem. Eng. J., 45 B19-B41. 

This time is characteristic of the relative motions of the small-scale fluid entities, adjacent to one another. These entities are produced by the fragmentation of the fluid phase under the effect of turbulence, and the relative motions serve to maintain spatial heterogeneity at small scales. Kolmogorov s theory, presented in Chapter 11, will help in quantifying fis , uk, and zk. ... 

A more comprehensive presentation of Kolmogorov s theory can be found in the book by M. Lesieur, Turbulence in Fluids (Springer-Verlag, 2007). 

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...

The starting point of Kolmogorov s theory is eqnation [11.1], which stipnlates that the rate of turbulent kinetic energy dissipation is independent of viscosity and hence depends only on Ums and It ... 

---