Turbulence eddy concept

In an interesting attempt to overcome the limitations found in the turbulent breakage models described above Martmez-Bazan et al [78] (see also Lasheras et al [58]) proposed an alternative model in the kinetic theory (microscopic) framework based on purely kinematic ideas to avoid the use of the incomplete turbulent eddy concept and the macroscopic model formulation. 

In the case of turbulent flow it is possible to employ the same eddy concepts as were used in connection with the consideration of thermal transport. In the case of steady, uniform flow between parallel plates the eddy diffusivity may be defined by... 

The Eddy Dissipation Concept (EDC) is used for treating the interaction between turbulence and chemistry in flames 12]. The method is based on a detailed description of the dissipation of turbulent eddies. In the EDC the total space is subdivided into a reaction space, called the fine structures and the surrounding fluid. In the presented reaction scheme the reactions Cl, C2, and C3 arc treated as taking place only in these fine structures, i.e. only on the smallest turbulent length scales. 

The two-equation models (especially, the k-s model) discussed above have been used to simulate a wide range of complex turbulent flows with adequate accuracy, for many engineering applications. However, the k-s model employs an isotropic description of turbulence and therefore may not be well suited to flows in which the anisotropy of turbulence significantly affects the mean flow. It is possible to encounter a boundary layer flow in which shear stress may vanish where the mean velocity gradient is nonzero and vice versa. This phenomenon cannot be predicted by the turbulent viscosity concept employed by the k-s model. In order to rectify this and some other limitations of eddy viscosity models, several models have been proposed to predict the turbulent or Reynolds stresses directly from their governing equations, without using the eddy viscosity concept. 

The eddy concept was thus introduced. An eddy still eludes precise definition, but in one interpretation it is conceived to be a turbulent motion, localized within a region of a certain size, that is at least moderately coherent over this region. The region of a large eddy can also contain smaller eddies. 

Sheldon started his Ph.D. studies at a time when the field of aerosol science was in its early stages of development. Working with H.F. Johnstone, he focused on how particles in turbulent airflow are deposited on the walls of pipes and ducts. Sheldon made important contributions right from the start he introduced the notion of a stopping distance of a particle injected into stagnant air, and then used this concept to predict particle motion through the viscous boundary layer to the surface. His thesis work laid the foundation for much of the later work on deposition of particles in industrial systems as well as dry deposition from the ambient atmosphere, where turbulent eddies impart velocities normal to the mean flow and enable particles to reach the surface. 

The above emulsification methods (perhaps except the Couette flow technique) have as a common feature that the final DSD is primarily determined by the interaction of turbulent eddies with interfaces. Note, however, that turbulence is hard to control and to maintain consistently throughout the whole reactor volume. From a practical point of view, it is almost impossible to predict the DSD after a scale-up based on laboratory-scale experiments. Emulsification techniques based on other principles are necessary to overcome these drawbacks. An alternative technique is the so-called membrane emulsification method where the liquid forming the disperse phase is pressed through a porous membrane. The other side of the membrane where the droplets are formed is in contact with the continuous phase. This concept is simple and it is assumed to be superior to the above techniques (35). The basic relationship of membrane emulsification (equation (8.10)) correlates the trans-membrane pressure required to start the drop-wise flow through the pores (ft) with the average pore diameter of the membrane (Dm) with being the contact angle of the mixture with the wall of the pore ... 

Mathematically, dispersion can be treated in the same manner as molecular diffusion, but the physical background is different dispersion is caused not only by molecular diffusion but also by turbulence effects. In flow systems, turbulent eddies are formed and they contribute to backmixing. Therefore, the operative concept of dispersion, the dispersion coefficient, consists principally of two contributions, that is, the one caused by molecular diffusion and the second one originating from turbulent eddies. Below we shall derive the RTD functions for the most simple dispersion model, namely, the axial dispersion model. 

Reynolds Stress Models. Eddy viscosity is a useful concept from a computational perspective, but it has questionable physical basis. Models employing eddy viscosity assume that the turbulence is isotropic, ie, u u = u u = and u[ u = u u = u[ = 0. Another limitation is that the... 

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

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). 

The (isotropic) eddy viscosity concept and the use of a k i model are known to be inappropriate in rotating and/or strongly 3-D flows (see, e.g., Wilcox, 1993). This issue will be addressed in more detail in Section IV. Some researchers prefer different models for the eddy viscosity, such as the k o> model (where o> denotes vorticity) that performs better in regions closer to walls. For this latter reason, the k-e model and the k-co model are often blended into the so-called Shear-Stress-Transport (SST) model (Menter, 1994) with the view of using these two models in those regions of the flow domain where they perform best. In spite of these objections, however, RANS simulations mostly exploit the eddy viscosity concept rather than the more delicate and time-consuming RSM turbulence model. They deliver simulation results of in many cases reasonable or sufficient accuracy in a cost-effective way. 

Concept (b) is less useful, except in rare cases where the energy spectrum has been measured. It is common to assume that the turbulence is homogeneous and isotropic and that the eddies in question are in the inertial ( — 5/3 power) subrange. This assumption is unlikely to be valid in an overall sense though it may be reasonable locally (GIO) or for the high wavenumber (small) eddies which are of primary interest. For an example of the application of the theory, see Middleman (Ml3). 

Axial dispersion. An axial (longitudinal) dispersion coefficient may be defined by analogy with Boussinesq s concept of eddy viscosity ". Thus both molecular diffusion and eddy diffusion due to local turbulence contribute to the overall dispersion coefficient or effective diffusivity in the direction of flow for the bed of solid. The moles of fluid per unit area and unit time an element of length 8z entering by longitudinal diffusion will be - D L (dY/dz)t, where D L is now the dispersion coefficient in the axial direction and has units ML T- (since the concentration gradient has units NM L ). The amount leaving the element will be -D l (dY/dz)2 + S2. The material balance equation will therefore be ... 

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