The Kolmogorov scale

Knowing the form of the Kolmogorov spectrum, the Kolmogorov scale can be readily calculated. Writing [11.18] and [11.19] for = 1/f , it follows from [11.16] that  

The quantities evaluated using equations [11.21]-[11.24] vary with the turbulent Reynolds number. Recall the result from Chapter 8 whereby Re should be greater 

3 By so doing, we can tacitly employ a representation based on logarithmic scales. 

Now that we know how to estimate the size fx of the smallest scales of turbulence, it is simple to conclude by summing up the principles that govern mixing and chemical reactions in a flow with homogenous turbirlence. Table 11.1 shows, for a chemical reaction, for micromixing (molecular diffusion), and for macromixing (dispersion at different scales), the characteristic times and the scales with which these various processes are associated. For a stirred reactor, the phenomenon involves six parameters of different nature  

Phenomenon Spatial scale Characteristic time Dimensionless numbers 

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In reactor design, it is very important to know how and where turbulence is generated and dissipated. In a liquid phase, it is also important that the smallest eddies are sufficiently small. The ratio between the reactor scale (I) and the smallest turbulent scale, the Kolmogorof scale rj), usually scales as L/x]aR . The Kolmogorov scale can also be estimated from the viscosity and the power dissipation T] = (v 30 xm in water with a power input of 1W kg and from the Bachelor scale 3 pm in liquids. For a liquid, the estimation of the time... 

The ratio of the Kolmogorov scale and the turbulence integral scale can be expressed in terms of the turbulence Reynolds number by... 

Two important length scales for describing turbulent mixing of an inert scalar are the scalar integral scale L, and the Batchelor scale A.B. The latter is defined in terms of the Kolmogorov scale r] and the Schmidt number by... 

Like the Kolmogorov scale in a turbulent flow, the Batchelor scale characterizes the smallest scalar eddies wherein molecular diffusion is balanced by turbulent mixing.3 In gas-phase flows, Sc 1, so that the smallest scales are of the same order of magnitude as the Kolmogorov scale, as illustrated in Fig. 3.1. In liquid-phase flows, Sc 1 so that the scalar field contains much more fine-scale structure than the velocity field, as... 

In a fully developed turbulent flow, the rate at which the size of a scalar eddy of length l,P decreases depends on its size relative to the turbulence integral scale L and the Kolmogorov scale ij. For scalar eddies in the inertial sub-range (ij < Ip, < Lu), the scalar mixing rate can be approximated by the inverse of the spectral transfer time scale defined in (2.68), p. 42 8... 

For scalar eddies smaller than the Kolmogorov scale, the physics of scalar mixing changes. As illustrated in Fig. 3.6, vortex stretching causes the scalar field to become one-dimensional at a constant rate (Batchelor 1959). Thus, for 1 < rj, the mixing rate can be approximated by /e i/2... 

This expression was derived originally by Batchelor (1959) under the assumption that the correlation time of the Kolmogorov-scale strain rate is large compared with the Kolmogorov time scale. Alternatively, Kraichnan (1968) derived a model spectrum of the form... 

Figure 3.13. Compensated scalar energy spectra at R-t = 500 normalized by the Kolmogorov scales. The compensated velocity energy spectrum is shown as a dotted line for comparison. The Schmidt numbers range from Sc = 10 2 to Sc = 102 in powers of 10.

Figure 4.1. Sketch of LES energy spectrum with the sharp-spectral filter. Note that all information about length scales near the Kolmogorov scale is lost after filtering. 

Reduction in size down to the Kolmogorov scale with no change in concentration at a rate that depends on the initial scalar length scale relative to the Kolmogorov scale. 

For fast equilibrium chemistry (Section 5.4), an equilibrium assumption allowed us to write the concentration of all chemical species in terms of the mixture-fraction vector c(x, t) = ceq( (x, 0). For a turbulent flow, it is important to note that the local micromixing rate (i.e., the instantaneous scalar dissipation rate) is a random variable. Thus, while the chemistry may be fast relative to the mean micromixing rate, at some points in a turbulent flow the instantaneous micromixing rate may be fast compared with the chemistry. This is made all the more important by the fact that fast reactions often take place in thin reaction-diffusion zones whose size may be smaller than the Kolmogorov scale. Hence, the local strain rate (micromixing rate) seen by the reaction surface may be as high as the local Kolmogorov-scale strain rate. 

Given the thinness of a diffusion flamelet, it is possible to neglect as a first approximation curvature effects, and to establish a local coordinate system centered at the reaction interface. By definition, X is chosen to be normal to the reaction surface. Furthermore, because the reaction zone is thin compared with the Kolmogorov scale, gradients with respect to X2 and X3 will be much smaller than gradients in the x direction (i.e., the curvature is small).112 Thus, as shown in Fig. 5.18, the scalar fields will be locally onedimensional. 

Flamelet wrinkling will be caused by vorticity, which is negligible below the Kolmogorov scale. 

Diffusion of momentum of the velocity fluctuations (or dissipation of turbulent kinetic energy) occurs at the Kolmogorov scale, which is estimated as... 

Taylor [159] stated that Xg can be regarded as a measure of the diameter of the smallest eddies which are responsible for the dissipation of turbulent energy. Pope ([121], p. 199) stated that this statement is incorrect, because it incorrectly supposes that Vrms is the characteristic velocity of the dissipative eddies. The characteristic length scale of the smallest eddies are the Kolmogorov scale, r/ = ( ) as will be further discussed shortly. 

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. 

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