Turbulence models, summar

In Section 2.2, the Reynolds-averaged Navier-Stokes (RANS) equations were derived. The resulting transport equations and unclosed terms are summarized in Table 2.4. In this section, the most widely used closures are reviewed. However, due to the large number of models that have been proposed, no attempt at completeness will be made. The reader interested in further background information and an in-depth discussion of the advantages and limitations of RANS turbulence models can consult any number of textbooks and review papers devoted to the topic. In this section, we will follow most closely the presentation by Pope (2000). 

A new term is introduced, the so-called Reynolds stresses m-m). The overbar denotes a time average. This term is the correlation between the turbulent velocity fluctuations and uj, and it describes the transport of momentum in the mean flow due to turbulence. This term is difficult to model, and over the years a variety of turbulence models have been developed. Turbulence models are necessary for calculating time-averaged flow fields directly, without first having to calculate a fully time-dependent flow field and then doing time averaging. The use of turbulence models is therefore much more computationally efficient. A detailed discussion is beyond the scope of this entry, but it is important to note that not all turbulence models are equally suited for all types of flow. Table 1 summarizes the most common turbulence models and their properties. 

Summarize the strengths and limitations of turbulence models and computational fluid dynamics (CFD) in general in the context of the design of mixing equipment. 

In a recent study Jakobsen et al. [71] examined the capabilities and limitations of a dynamic 2D axi-symmetric two-fluid model for simulating cylindrical bubble column reactor flows. In their in-house code all the relevant force terms consisting of the steady drag, bulk lift, added mass, turbulence dispersion and wall lift were considered. Sensitivity studies disregarding one of the secondary forces like lift, added mass and turbulent dispersion at the time in otherwise equivalent simulations were performed. Additional simulations were run with three different turbulence closures for the liquid phase, and no shear stress terms for the gas phase. A standard k — e model [95] was used to examine the effect of shear induced turbulence, case (a). In an alternative case (b), both shear- and bubble induced turbulence were accounted for by linearly superposing the turbulent viscosities obtained from the A — e model and the model of Sato and Sekoguchi [138]. A third approach, case (c), is similar to case (b) in that both shear and bubble induce turbulence contributions are considered. However, in this model formulation, case (c), the bubble induced turbulence contribution was included through an extra source term in the turbulence model equations [64, 67, 71]. The relevant theory is summarized in Sect. 8.4.4. 

The relationship between the CRE approach and the FM approach to modeling turbulent reacting flows is summarized in Table 1.1. Despite the obvious and significant differences... 

For convenience, the turbulence statistics used in engineering calculations of inhomogeneous, high-Reynolds-number turbulent flows are summarized in Table 2.4 along with the unclosed terms that appear in their transport equations. Models for the unclosed terms are discussed in Chapter 4. 

The gradient-flux model to describe turbulent diffusion (Eq. 18-70) has the disadvantage that turbulent diffusivity, Ex, is scale dependent. As discussed in more detail in Chapter 22, in natural systems Ex increases with increasing horizontal scale of diffusion. This means that the speed with which two fluid parcels are separated by turbulence increases the further they are from each other. This is because turbulent structures (eddies) of increasing size become effective when the size of a diffusing patch becomes larger. Typical ranges of turbulent diffusivities in the environment are summarized in Table 18.4. 

First, recall that the nondimensional Damkohler number, Da (Eq. 22-11 b), allows us to decide whether advection is relevant relative to the influence of diffusion and reaction. As summarized in Fig. 22.3, if Da 1, advection can be neglected (in vertical models this is often the case). Second, if advection is not relevant, we can decide whether mixing by diffusion is fast enough to eliminate all spatial concentration differences that may result from various reaction processes in the system (see the case of photolysis of phenanthrene in a lake sketched in Fig. 21.2). To this end, the relevant expression is L (kr / Ez)1 2, where L is the vertical extension of the system, Ez the vertical turbulent diffusivity, and A, the first-order reaction rate constant (Eq. 22-13). If this number is much smaller than 1, that is, if... 

In the previous section, stability criteria were obtained for gas-hquid bubble columns, gas-solid fluidized beds, liquid-sohd fluidized beds, and three-phase fluidized beds. Before we begin the review of previous work, let us summarize the parameters that are important for the fluid mechanical description of multiphase systems. The first and foremost is the dispersion coefficient. During the derivation of equations of continuity and motion for multiphase turbulent dispersions, correlation terms such as esv appeared [Eqs. (3) and (10)]. These terms were modeled according to the Boussinesq hypothesis [Eq. (4)], and thus the dispersion coefficients for the sohd phase and hquid phase appear in the final forms of equation of continuity and motion [Eqs. (5), (6), (14), and (15)]. However, for the creeping flow regime, the dispersion term is obviously not important. 

These transport equations contain four empirical parameters, which are listed in Table 3.1 along with the parameter appearing in Eq. (3.20). The values of these parameters are obtained with the help of experimental information about simple flows such as decay of turbulence behind the grid (Launder and Spalding, 1972). Before discussing the modifications to the standard k-s model and its recent renormalization group version, it will be useful to summarize implicit and explicit assumptions underlying the k- model ... 

We summarize a number of simulations aimed at deciphering some of the basic effects which arise from the interaction of chemical kinetics and fluid dynamics in the ignition and propagation of detonations in gas phase materials. The studies presented have used one- and two-dimensional numerical models which couple a description of the fluid dynamics to descriptions of the detailed chemical kinetics and physical diffusion processes. We briefly describe, in order of complexity, a) chemical-acoustic coupling, b) hot spot formation, ignition and the shock-to-detonation transition, c) kinetic factors in detonation cell sizes, and d) flame acceleration and the transition to turbulence. 

The standard k-s model, as presented by Launder and Spalding [2], is by far the most widely-used two-equation eddy viscosity model, also for modeling turbulence in stirred tank reactors. The popularity of the model and its wide use and testing has thrown light on both its capabilities and its shortcomings, which are well documented in the literature [2-8]. For high turbulent Reynolds numbers, the model may be summarized as follows ... 

The classical model of a normal turbulent jet was recently extended to a S3mthetic jet [6, 7]. This is summarized in this section. Axisymmetric free turbulent jets can be solved analytically by using the Prandtl mixing-length model. The model findings are summarized by... 

The accuracy of the Gaussian diffusion model has been reviewed in a note prepared by the American Meteorological Society 1977 Committee on Atmospheric Turbulence and Diffusion. The Committee estimates can be summarized as follows ... 

The purpose of this chapter is to show the role of the model used for the interfacial shear at different stages of stability analyses of the stratified flow configuration and to summarize progress made in formulating a closure law which reflects the dynamics of the interaction involved in turbulent gas flow over a mobile wavy interface. 

The laws of convection and conduction are contained directly in Eq. (25.8), whereas radiation requires an additional radiation model and is taken into account using the energy source density ai or an additional source term. The terms keff and describe the effective conductivity and effective stress tensor, respectively, and comprise a laminar and a turbulent part. The other production and applied densities are summarized in a simpHfied form in the energy source term St, in Eq. (25.8). 

To summarize the solution process for the k-e model, transport equations are solved for the turbulent kinetic energy and dissipation rate. The solutions for k and 8 are used to compute the turbulent viscosity, Xf Using the results for Xt and k, the Reynolds stresses can be computed from the Boussinesq hypothesis for substitution into the momentum equations. Once the momentum equations have been solved, the new velocity components are used to update the turbulence generation term, Gk, and the process is repeated. 

Abstract Computational heat transfer (CHT) should be included in the computational mass transfer (CMT) model system if thermal effect is involved in the simulated process. In this chapter, as a preparatory material parallel to Chap. 1, the CHT method for turbulent flow is summarized. This chapter focuses on the closure of the time-averaged energy equation. The unknown term to be solved is the covariant composed of the velocity and temperature fluctuations. Two modeling methods for this term are introduced, namely the two-equation method... 

The flow pattern of the vortex motion of the gas in reverse-flow cyclone is quite complex. First, it is three-dimensional second, the flow is turbulent An exact analysis is therefore difficult Soo (1989) has summarized a fundamental analysis of velocity profiles and pressure drops in such a cyclone. He has also analyzed the governing particle diffusion equation in the presence of electrostatic, gravitational and centrifugal forces. He has then provided an analytical expression for partide collection efficiency under a number of limiting conditions. We wiU, however, opt here for a much simpler model of particle separation in a cyclone developed by Clift et id. (1991). This approach is based on a modification of the original model by Leith and Licht (1972). The model will be... 

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