Length scales Kolmogorov

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... 

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... 

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

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 .

A-k Kolmogorov length scale where inertial sub-range ends... 

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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