Kolmogorov scale

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... 

Quantity Integral scale Taylor scale Kolmogorov scale... 

By definition, the dissipation range is dominated by viscous dissipation of Kolmogorov-scale vortices. The characteristic time scale rst in (2.74) can thus be taken as proportional to the Kolmogorov time scale rn, and taken out of the integral. This leads to the final form for (2.70),... 

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. 

Since the molecular diffusivities are used in (5.254), the interval length L(t) and the initial conditions will control the rate of molecular diffusion and, subsequently, the rate of chemical reaction. In order to simulate scalar-gradient amplification due to Kolmogorov-scale mixing (i.e., for 1 < Sc), the interval length is assumed to decrease at a constant rate ... 

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. 

Dfc Damkohler number characterizing Kolmogorov-scale fluctuations D Damkohler number characterizing large-scale fluctuations I integral length scale... 

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... 

Computational fluid dynamics enables us to investigate the time-dependent behavior of what happens inside a reactor with spatial resolution from the micro to the reactor scale. That is to say, CFD in itself allows a multi-scale description of chemical reactors. To this end, for single-phase flow, the space resolution of the CFD model should go down to the scales of the smallest dissipative eddies (Kolmogorov scales) (Pope, 2000), which is inversely proportional to Re-3/4 and of the orders of magnitude of microns to millimeters for typical reactors. On such scales, the Navier-Stokes (NS) equations can be expected to apply directly to predict the hydrodynamics of well-defined system, resolving all the meso-scale structures. That is the merit of the so-called DNS. 

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