Kolmogorov flows

For turbulent fluid-indueed stresses aeting on partieles it is neeessary to eon-sider the strueture and seale of turbulenee in relation to partiele motion in the flow field. There is as yet, however, no eompletely satisfaetory theory of turbulent flow, but a great deal has been aehieved based on the theory of isotropie turbulenee (Kolmogorov, 1941). 

By its random nature, turbulence does not lend itself easily to modelling starting from the differential equations for fluid flow (Navier-Stokes). However, a remarkably successful statistical model due to Kolmogorov has proven very useful for modelling the optical effects of the atmosphere. 

The Kolmogorov velocity field mixes packets of air with different passive scalars a passive scalar being one which does not exchange energy with the turbulent velocity flow. (Potential) temperature is such a passive scalar and the temperature fluctuations also follow the Kolmogorov law with a different proportionality constant... 

The cornerstone of LES methodology is the self-similarity theory of Kolmogorov stating that even though the large structures of a turbulenf flow depend on fhe boundary and initial conditions, the finer scales have a universal... 

The validity of Eqs. (3-5) are bond on the condition of fully developed turbulent flow which only exists if the macro turbulence is not influenced by the viscosity. This is the case if the macro turbulence is clearly separated from the dissipation range by the inertial range. This is given if the macro scale A is large in comparison to Kolmogorov s micro scale qp Liepe [1] and Mockel [24] found out by measurement of turbulence spectra s the following condition ... 

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 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... 

Sleiched278 has indicated that this expression is not valid for pipe flows. In pipe flows, droplet breakup is governed by surface tension forces, velocity fluctuations, pressure fluctuations, and steep velocity gradients. Sevik and Park 279 modified the hypothesis of Kolmogorov, 280 and Hinze, 270 and suggested that resonance may cause droplet breakup in turbulent flows if the characteristic turbulence frequency equals to the lowest or natural frequency mode of an... 

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... 

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... 

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. 

Thus, the criterion to be satisfied if a laminar flame is to exist in a turbulent flow is that the laminar flame thickness <5L be less than the Kolmogorov microscale 4 of the turbulence. 

Liu and Neeld (78) used VisiMix software to calculate shear rates in laboratory, pilot plant, and production scale vessels. Their results (Table 3) showed marked differences, by as much as two orders of magnitude, in the shear rates calculated in the conventional manner [from tip speed and the distance from impeller tip to baffle, i.e., y = ND/ T — Z))] and the shear rates computed by VisiMix. The latter s markedly higher shear rates resulted from VisiMix s definition of the shear rate in terms of Kolmogorov s model of turbulence and the distribution of flow velocities. Note that VisiMix s estimates of the respective shear rates in the vicinity of the impeller blade are comparable at all scales while the shear rates in the bulk volume or near... 

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