Kolmogorovs theory

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. 

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  

Consequent to this fact, it is deduced that dissipation occurs at small scales. It is easy to evaluate the order of magnitude of the different terms in Navier-Stokes equation, for any scale I inside the interval covered by the turbulent energy spectmm i it). By associating velocity ui with length t, the orders of magnitude of the various terms in Navier-Stokes eqnation can be evaluated  

We thus find that the dissipation of tnrbulent kinetic energy is negligible for those scales I for which (M/f)/v l, whereas it is dominant at scales such that 

Such a formulation ejqtresses the simple idea that transfer at any scale t within the inertial zone depends only on the characteristics of turbulence at that scale. The turbulent stracture can only transfer energy to a smaller scale to the extent of the energy uf it contains and with its own characteristic time. That is the reason for bringing in the characteristic time 

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Middleman, S., 1965. Mass transfer from particles in agitated systems Application of the Kolmogorov theory. American Institute of Chemical Engineers Journal, 11, 750-752. 

The hydrogen absorption/desorption kinetics are usually analyzed by applying the JMAK (Johnson-Mehl-Avrami-Kolmogorov) theory of phase transformations, which is based on nucleation and growth events [166-168] where a is the fraction transformed at time t or alternatively for hydrides the fraction absorbed... 

When the turbulence is not too large, the main force on the droplets is caused by the shear imposed by the surrounding eddies. With the help of the Kolmogorov theory for turbulent flow, the external, disrupting force is can estimated as Text = V(e /c)- We then end up with a definition of the critical Weber number ... 

The energy dissipation number is derived from the Kolmogorov theory [289, 290]. This theory is based on the concept, that in the turbulent flow range the kinetic energy is transferred by inertial forces from large to ever smaller eddies, until it is finally dissipated through viscosity forces. The eddies produced by the stirrer have the size of the stirrer head and are responsible for the macroscopic turbulence, the smallest eddies on the other hand are directionless. On the microscale so-called isotropic turbulence exists, see Section 1.4.2. 

With increasing the droplets of the dispersed phase damp the turbulence. On the other hand droplet coalescence increases in the turbulence-deficient flow regions in the stirred tank. The result is an increase in the average droplet size. Doulah s [103] deliberations based on the Kolmogorov theory led to the following relationship for... 

If Redr < 1, the flow near a drop is laminar, and it would be broken up in the regime TV. Now the local flow velocity is needed at the scale of a droplet d [i.e., u (d)], and this scale is (much) smaller than /e. According to the Kolmogorov theory the velocity is given by... 

The dependence of the Reynolds number on the local velocity can be established in terms of the Kolmogorov theory of turbulence. 

Some results are given in Table 2.3. It should be remembered that eqn. (2.16), like all equations in Kolmogorov theory, has scaling relations. Often, eqn. (2.16) is also applied for the average droplet size resulting from emulsification, say d - In that case, the proportionality constant would be smaller than for [Pg.67]

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