Length scales Kolmogorov scale

Most of the energy dissipation occurs on a length scale about 5 times the Kolmogorov eddy size. The characteristic fluctuating velocity for these energy-dissipating eddies is about 1.7 times the Kolmogorov velocity. 

Rate of turbulence production (m s ) Velocity of a turbulent eddy of size X Rate of turbulence dissipation (m s ) Kolmogorov length scale (m)... 

Cherry and Papoutsakis [33] refer to the exposure to the collision between microcarriers and influence of turbulent eddies. Three different flow regions were defined bulk turbulent flow, bulk laminar flow and boundary-layer flow. They postulate the primary mechanism coming from direct interactions between microcarriers and turbulent eddies. Microcarriers are small beads of several hundred micrometers diameter. Eddies of the size of the microcarrier or smaller may cause high shear stresses on the cells. The size of the smallest eddies can be estimated by the Kolmogorov length scale L, as given by... 

In stirred chemical reactors, unlike in combustion and with other gas-phase reactions, these closure terms should take into account that for liquids the Schmidt number (Sc = v/D) is in the order 100-1,000, and, hence, the role of species diffusion at scales within the Kolmogorov eddies should explicitly be taken into account (Kresta and Brodkey, 2004). Essential is that diffusion of chemical species is governed by the Batchelor length scale rjB which obeys to... 

An alternative approach (e.g., Patterson, 1985 Ranade, 2002) is the Eulerian type of simulation that makes use of a CDR equation—see Eq. (13)—for each of the chemical species involved. While resolution of the turbulent flow down to the Kolmogorov length scale already is far beyond computational capabilities, one certainly has to revert to modeling the species transport in liquid systems in which the Batchelor length scale is smaller than the Kolmogorov length scale by at least one order of magnitude see Eq. (14). Hence, both in RANS simulations and in LES, species concentrations and temperature still fluctuate within a computational cell. Consequently, the description of chemical reactions and the transport of heat and species in a chemical reactor ask for subtle approaches as to the SGS fluctuations. 

In the normalized energy spectrum, k— 1 corresponds to the inverse of the local integral length scale and k — Re 4 to the inverse of the local Kolmogorov length scale. The range of wavenumbers in Fig. 1 over which the slope is —5/3 is... 

In this definition, ps and pt are the solid and fluid densities, respectively. The characteristic diameter of the particles is ds (which is used in calculating the projected cross-sectional area of particle in the direction of the flow in the drag law). The kinematic viscosity of the fluid is vf and y is a characteristic strain rate for the flow. In a turbulent flow, y can be approximated by l/r when ds is smaller than the Kolmogorov length scale r. (Unless the turbulence is extremely intense, this will usually be the case for fine particles.) Based on the Stokes... 

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... 

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... 

The time step required for accurate solutions of (4.3) is limited by the need to resolve the shortest time scales in the flow. In Chapter 3, we saw that the smallest eddies in a homogeneous turbulent flow can be characterized by the Kolmogorov length and time scales. Thus, the time step h must satisfy3 /V l/2... 

Likewise, good spatial accuracy requires that the maximum wavenumber (K) be inversely proportional to the Kolmogorov length scale ... 

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 ... 

Figure 5.17. A laminar diffusion flamelet occurs between two regions of unmixed fluid. On one side, the mixture fraction is unity, and on the other side it is null. If the reaction rate is localized near the stoichiometric value of the mixture fraction st, then the reaction will be confined to a thin reaction zone that is small compared with the Kolmogorov length scale.

Figure 5.18. The diffusion flamelet can be approximated by a one-dimensional transport equation that describes the change in the direction normal to the stoichiometric surface. The rate of change in the tangent direction is assumed to be negligible since the flamelet thickness is small compared with the Kolmogorov length scale. The flamelet approximation is valid when the reaction separates regions of unmixed fluid. Thus, the boundary conditions on each side are known, and can be uniquely expressed in terms of .

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

Deduced in 1941 by A. N. Kolmogorov, it is generally referred to the Kolmogorov length or dissipation scale (9). 

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