Turbulent transport, models

The commercial software ANSYS FLUENT was used for CFD analysis. Unsteady flow was modeled to obtain the variation of granule temperature with time. The k-to with shear-stress transport turbulence model was used. The modeled flue gases included four gas species, referring to CO2, H2O, N2, and O2. The boundary condition was velocity-inlet at the inlet, pressure outlet at the outlet, and symmetry for the four sides. Second-order upwind scheme was used for the momentum, species, and energy equations. 

The two-fluid method using the volume of fluid (VOF) approach, coupled with the k-co SST (shear stress transport) turbulence model, is employed to simulate the formation and fragmentation of a swirling conical liquid sheet firom a typical swirl nozzle. In the VOF approach, the volume firaction of liquid (/) is defined in each computational cell. If the cell is completely filled with liquid then/= 1 and if it is filled with gas then its value is 0. At the gas-liquid interface, the value of / is between 0 and 1. In the OpenFOAM VOF formulation, the interface motion is governed by the convective transport equation of the volume fraction... 

HOTM AC/RAPTAD contains individual codes HOTMAC (Higher Order Turbulence Model for Atmospheric Circulation), RAPTAD (Random Particle Transport and Diffusion), and computer modules HOTPLT, RAPLOT, and CONPLT for displaying the results of the ctdculalinns. HOTMAC uses 3-dimensional, time-dependent conservation equations to describe wind, lempcrature, moisture, turbulence length, and turbulent kinetic energy. 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

These two transport equations for k and e form an inherent part of any k i model of RANS-simulations. As the result of closing the turbulence modeling such that no further unknown variables and equations are introduced, the e-equation does contain some terms that are still the result of modeling, albeit at the very small scales (e.g., Rodi, 1984). 

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. 

RANS simulations usually exploiting some k-e turbulence model, intended for global information on the average flow field and the global transport phenomena in full-scale process equipment, with additional output (of limited confidence level) on spatial distributions of k and e ... 

The left-hand sides of Eqs. (25)-(29) have the same form as Eq. (5) and represent accumulation and convection. The terms on the right-hand side can be divided into spatial transport due to diffusion and source terms. The diffusion terms have a molecular component (i.e., /i and D), and turbulent components. We should note here that the turbulence models used in Eqs. (26) and (27) do not contain corrections for low Reynolds numbers and, hence, the molecular-diffusion components will be negligible when the model is applied to high-Reynolds-number flows. The turbulent viscosity is defined using a closure such as... 

In order to model turbulent reacting flows accurately, an accurate model for turbulent transport is required. In Chapter 41 provide a short introduction to selected computational models for non-reacting turbulent flows. Here again, the goal is to familiarize the reader with the various options, and to collect the most important models in one place for future reference. For an in-depth discussion of the physical basis of the models, the reader is referred to Pope (2000). Likewise, practical advice on choosing a particular turbulence model can be found in Wilcox (1993). 

For canonical turbulent flows (Pope 2000), the flow parameters required to complete the CRE models are readily available. However, for the complex flow fields present in most chemical reactors, the flow parameters must be found either empirically or by solving a CFD turbulence model. If the latter course is taken, the next logical step would be to attempt to reformulate the CRE model in terms of a set of transport equations that can be added to the CFD model. The principal complication encountered when following this path is the fact that the CRE models are expressed in a Lagrangian framework, whilst the CFD models are expressed in an Eulerian framework. One of the main goals of this book... 

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. 

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). 

The next level of turbulence models introduces a transport equation to describe the variation of the turbulent viscosity throughout the flow domain. The simplest models in this category are the so-called one-equation models wherein the turbulent viscosity is modeled by... 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

Transported PDF methods are continuing to develop in various directions (i.e, compressible flows, FES turbulence models, etc.). A detailed overview of transported PDF methods is presented in the following two chapters. 

The transported PDF equation contains more information than an RANS turbulence model, and can be used to derive the latter. We give two example derivations U) and (uuT below, but the same procedure can be carried out to find any one-point statistic of the velocity and/or composition fields.25... 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

The transported PDF models discussed so far in this chapter involve the velocity and/or compositions as random variables. In order to include additional physics, other random variables such as acceleration, turbulence dissipation, scalar dissipation, etc., can be added. Examples of higher-order models developed to describe the turbulent velocity field can be found in Pope (2000), Pope (2002a), and Pope (2003). Here, we will limit our discussion to higher-order models that affect the scalar fields. 

The Reynolds stress model requires the solution of transport equations for each of the Reynolds stress components as well as for dissipation transport without the necessity to calculate an isotropic turbulent viscosity field. The Reynolds stress turbulence model yield an accurate prediction on swirl flow pattern, axial velocity, tangential velocity and pressure drop on cyclone simulation [7,6,13,10],... 

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