Kolmogorov-scale turbulence

Although vortices of small scale, such as Kolmogorov scale or Taylor microscale, are significant in modeling turbulent combustion [4,6-9], vortices of large scale, in fhe order of millimeters, have been used in various experiments to determine the flame speed along a vorfex axis. 

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 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. 

Figure 25.1 Regimes of turbulent combustion 1 — offshore flares, 2 — spark-ignition engines, 3 — supersonic combustion, Kl — turbulent kinetic energy referred to laminar ratio of kinematic viscocity to chemical time, — Damkohler number based on Kolmogorov scale, Ld — integral scale referred to thickness of laminar deflagration...

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

FIGURE 10.5. Parameter plane of nondimensional intensity and nondimensional Kolmogorov scale of turbulence for premixed turbulent combustion, showing regimes of combustion and lines for constant values of turbulence Reynolds numbers, nondimensional integral scale, and a Damkohler number. 

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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