Isotropy, turbulent

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). 

It is then assumed that due to this separation in scales, the so-called subgrid scale (SGS) modeling is largely geometry independent because of the universal behavior of turbulence at the small scales. The SGS eddies are therefore more close to the ideal concept of isotropy (according to which the intensity of the fluctuations and their length scale are independent of direction) and, hence, are more susceptible to the application of Boussinesq s concept of turbulent viscosity (see page 163). 

Phase-averaged values of 4 in a plane midway between two baffles of a stirred tank have been plotted in Fig. 1 (from Hartmann et al., 2004a) for two different SGS models (Smagorinsky and Voke, respectively) in LES carried out in a LB approach. The highest values, i.e., the strongest deviations from isotropy, occur in the impeller zone, in the boundary layers along wall and bottom of the tank, and at the separation points at the vessel wall from which the anisotropy is advected into the bulk flow. In the recirculation loops, the turbulent flow is more or less isotropic. 

As this chapter is primarily concerned with single-drop performance, it seems best to omit consideration of drop sizes in highly turbulent liquid fields. The work of Shinnar and Church (S7), utilizing Kolmogo-roff s hypothesis of local isotropy, seems to bear excellent promise from a fundamental viewpoint. Correlating equations for predicting drop size in stirred tanks and mixers have been given by Treybal (T3). 

Mellor and Herring also examine MRS closures, and show how the MTEN closure results from the MRS equations with the additional assumption of small departures from isotropy. While this approach is academically interesting, even the most weakly strained flows are far from isotropic (C4), and hence the main selling point for MTE methods is that they work very well for predicting a wide class of turbulent shear flows. Examples are given in the following section. 

Examination of Eq. (22) shows that the R, equations contain a pressure-strain-rate correlation term that vanishes in the contraction [Eq. (23)]. The effect of this term must therefore be to transfer energy conservatively between the three components Rn, R22, and R33, and it is generally believed that this transfer tends to produce isotropy in the turbulent motions. Modelings of this term should incorporate this feature. A plausible model of this term, supported somewhat by the data of Champagne et al. (C4) is... 

Homogeneous isotropic turbulence is a mathematical idealization of real turbulence, that was introduced by [159], allowing us to simplify the analysis of turbulence considerably, and thus gain insight into its behavior. Nevertheless, real turbulent flows do rarely approach homogeneity and isotropy. However, by using the homogeneous and isotropic turbulence concept the mathematical problem simplifies considerably and it is possible to obtain many specific mathematical results which explain several aspects of turbulent flows. 

In the following we will consider some basic results of the statistics of such a homogeneous isotropic turbulent field. The consequences of homogeneity and isotropy for the correlation functions were worked out by von Karman and Howarth [179] and the full derivations are available in classical books like [66, 8, 112, 113]. 

Approximately stated Kolmogorov s hypothesis of local isotropy yields ([83] see also [121], p. 184) At sufficiently high Reynolds number, the small scale turbulent motions are statistically isotropic. 

In the considered case, the basic mechanisms of formation of droplets in the turbulent gas flow are processes of coagulation and breakage of drops. These two processes proceed simultaneously. As a result, the size distribution of the drops is established. Assuming homogeneity and isotropy of the turbulent flow, this distribution looks like a logarithmic normal distribution [1] ... 

The form of Eq. (5.9) models a retum-to-isotropy effect due to fluctuating interfacial momentum coupling and reduces the turbulent viscosity from that predicted by the single-phase model. The turbulence energy exchange rate coefficient Ey is given by... 

Laushey, L.M. (1951). Momentum and kinetic energy of turbulence, dispersion symmetry, and isotropy of the fluctuations. PhD Thesis. Carnegie Institute of Technology Pittsburgh. 

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