Kolmogorov subrange

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. 

According to Kolmogorov s (1941) inertial subrange theory of turbulence, the exponent in Eq. 1 is m = 3. Inserting into Eq. 4 ... 

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). 

This model further assumes that the size of the parent particles is in the inertial subrange of turbulence. Therefore, it implies that dmin < d < dmax provided that dmin > Ad, where Ad is the Kolmogorov length scale of the underlying turbulence. Otherwise, dmin is taken to be equal to Ad. However, no assumption needs to be made about the minimum and maximum eddy size that can cause particle breakage. All eddies with sizes between the Kolmogorov scale and the integral scale are taken into account. 

Where e is the rate of kinetic energy dissipation per unit mass and Cx is of order 1. Equation (36) is valid for drops whose diameter falls within the inertial subrange of turbulence, < d < L, where L is the integral scale of turbulence and q = is the Kolmogorov microscale of turbulence. 

The method proposed by Kolmogorov (53) and Hinze (54) was extended by Ba+dyga and Podgorska (56) and Ba+dyga and Bourne (57) to the case of more realistic intermittent turbulence, which was described by means of multifractal formalism. In this model drop size in the inertial subrange also depends on the integral scale of turbulence, which is related to the scale of the system. This new formalism predicts... 

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... 

For the second subrange, viscous subrange, Kolmogorov suggested that... 

Other inadequacies of Kolmogorov s theory in the manner of its use by Shinnar and Church (1960) and Brian et al. (1969) to correlate solid-liquid mass transfer in stirred reactors have been dealt in some detail by Levins and Glastonbury (1972a). Stewart and Townsend (1951) have estimated the value of Re to be 1500 for the existence of the inertial subrange. Taylor (1935) estimated that for the region outside the impeller, the value of ROj. for conditions typical of stirred reactors is -144. This is much smaller than that required by Stewart and Townsend s estimated value of 1500. Therefore, the validity of Equation 6.8 outside the impeller discharge stream is doubtful. Further, the use of in Equation 6.10 for Re is not justified by... 

Turbulence occurs in any sufficiently rapid flow when the fluid inertia exceeds its molecular friction. It is the typical flow regime when suspensions of relatively low viscosity are dispersed—e.g. by stirring or in nozzles. Turbulent flow is characterised by multiscale eddy structures that erratically move through the flow field and cause local fluctuations in velocity and pressure. The velocity fluctuations can be quantified by the effective velocity difference Am over a distance Ar (Kolmogorov 1958). For the inertial subrange of microturbulence, it amounts to ... 

When a particle is so small that Tps/< 1, it is capable of responding to the dynamic world of the Kolmogorov eddies while obeying Stokes law. A larger particle in the inertial subrange, for which Tt < Tp < Tf, is too big to respond to eddies of size smaller than and its motion wiU (mainly)... 

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