RSM, turbulence model

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. 

The Fluent code with the RSM turbulence model, predict very well the pressure drop in cyclones and can be used in cyclone design for any operational conditions (Figs. 3, 5, 7 and 8). In the CFD numerical calculations a very small pressme drop deviation were observed, with less than 3% of deviation at different inlet velocity which probably in the same magnitude of the experimental error. The CFD simulations with RNG k-e turbulence model still yield a reasonably good prediction (Figs. 3, 5, 7 and 8) with the deviation about 14-20% of an experimental data. It considerably tolerable since the RNG k-e model is much less on computational time required compared to the complicated RSM tmbulence model. In all cases of the simulation the RNG k-< model considerably underestimates the cyclone pressme drop as revealed by Griffiths and Boysan [8], However under extreme temperature (>850 K) there is no significant difference between RNG k-< and RSM model prediction. 

Stress Model (RSM) turbulence model was one of the pioneering attempts to simulate a dust-laden swirled flow inside a cyclone. 

An appropriate model of the Reynolds stress tensor is vital for an accurate prediction of the fluid flow in cyclones, and this also affects the particle flow simulations. This is because the highly rotating fluid flow produces a. strong nonisotropy in the turbulent structure that causes some of the most popular turbulence models, such as the standard k-e turbulence model, to produce inaccurate predictions of the fluid flow. The Reynolds stress models (RSMs) perform much better, but one of the major drawbacks of these methods is their very complex formulation, which often makes it difficult to both implement the method and obtain convergence. The renormalization group (RNG) turbulence model has been employed by some researchers for the fluid flow in cyclones, and some reasonably good predictions have been obtained for the fluid flow. 

The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k-e model, the RSM involves five more differential equations). The number of equations further depends on the character of the simulation whether it is 3-D, 21/2-D, or just 2-D (see below, under The domain and the grid ). In the case of two-phase flow simulations, the use of two-fluid models implies doubling the number of NS equations required for single-phase flow. All this may urge the development of more efficient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. 

All of these models require some form of empirical input information, which implies that they are not general applicable to any type of turbulent flow problem. However, in general it can be stated that the most complex models such as the ASM and RSM models offer the greatest predictive power. Many of the older turbulence models are based on Boussinesq s (1877) eddy-viscosity concept, which assumes that, in analogy with the viscous stresses in laminar flows, the Reynolds stresses are proportional to the gradients of the time-averaged velocity components ... 

Computational experience has revealed that the two-equation models, employing transport equations for the velocity and length scales of the fluctuating motion, often offer the best compromise between width of application and computational economy. There are, however, certain types of flows where the k-e model fails, such as complex swirling flows, and in such situations more advanced turbulence models (ASM or RSM) are required that do not involve the eddy-viscosity concept (Launder, 1991). According to the ASM and the RSM the six components of the Reynolds stress tensor are obtained from a complete set of algebraic equations and a complete set of transport equations. These models are conceptually superior with respect to the older turbulence models such as the k-e model but computationally they are also (much) more involved. 

Turbulence models. Do we have to consider turbulent flow (in complex) geometries or not Especially in systems with complex geometries where turbulent flows prevail, simple turbulence models may fail and in such circumstances advanced turbulence models (ASM or RSM) are required. 

In studies that involve the CFD analysis of turbulent fluid flow, the k-t model is most frequently used because it offers the best compromise between width of application and computational economy (Launder, 1991). Despite its widespread popularity the k-e model, if used to generate an isotropic turbulent viscosity, is inappropriate for simulation of turbulent swirling flows as encountered in process equipment such as cyclones and hydrocyclones (Hargreaves and Silvester, 1990) and more advanced turbulence models such as the ASM or the RSM should be considered. Because these models are computationally much more demanding and involve an increased number of empirical parameters compared to the k-e model, other strategies have been worked out (Boysan et al, 1982 Hargreaves and Silvester, 1990) to avoid the isotropic nature of the classical k-e model. 

A 2-D CFD model has been set up using FLUENT4.5 in a joint EU JOULE project with FLUENT and ALSTOM. As turbulence models the k- model and Reynolds Stress model (RSM) have been applied. As chemistry models a chemical equilibrium model has been applied and on the other hand two models describing finite reaction chemistry, i.e. the laminar flamelet model and the reaction progress variable model. The comparison between experiments and the numerical results from the three chemistry models show that the chemical equilibrium model is sufficient to predict the combustion of LCV gas at elevated pressures, since deviation from chemical equilibrium is small due to the fast reactions. Hence no improvements are expected and have been observed from kinetically limited models. The RSM with constants Cl and C2 in the pressure-strain term proposed by Gibson and Younis [17] seems to yield the best predictions, however, the influence of the type of turbulence model (RSM or k- e) on the species concentrations and temperature predictions is not very large. 

Reynolds Stress Models (RSM) Most general model of all classical turbulence models Performs well for many complex flows including non-circular ducts and curved flows Computationally expensive (seven extra PDEs) Performs as poorly as k-s in some flows due to problems with s equation Not widely validated... 

Compared with Equation 4.6, Equation 4.12 contains the term -pUi Uj, the so-called Reynolds stress, which represents the effect of turbulence and must be modeled by the CFD code. Limited computational resources restrict the direct simulation of these fluctuations, at least for the moment. Therefore the transport equations are commonly modified to account for the averaged fluctuating velocity components. Three commonly applied turbulence modeling approaches have been used in the CFD model of spray drying system, i.e., k-Q model (Launder and Spalding 1972, 1974), RNG k-e model (Yakhot and Orszag 1986), and a Reynolds stress model (RSM) (Launder et al. 1975). 

These contain what is known as Reynolds stress R as an additional term describing turbulence. To complete the system of equations, Reynolds stress is usually determined by an heuristic turbulence model. To do so, FLUENT provides one-equation models (Sparlat-Allmaras), two-equation models k—e model, k—a> model), and the closure approaches k—kl—a> transition model, SST transition model, v —f model, and the Reynolds stress model (RSM) [32, 36]. A two-equation model widely used in practice is the standard k—e model, which is a turbulence viscosity model and represents a good compromise between accuracy and computational cost The flow turbulence is described here by the turbulent kinetic energy k and its degree of dissipation e. In accordance with Eq. (25.1), these two variables are determined using two additional conservation equations. 

Currently the widely used turbulence models are standard K-s model, RNG K-e model and the Reynolds stress model (RSM). Standard K-s model is based on isotropic turbulence model, its simulation result error of separator flow field is large (Shan Yongbo, 2005). RNG K-s model has improved with a standard K-s model, but there are still larger defects. To improve the cyclone vortex field strength prediction results a greater extent, algebraic stress turbulence model based... 

Eigure 20 compares the predictions of the k-Q, RSM, and ASM models and experimental data for the growth of the layer width 5 and the variation of the maximum turbulent kinetic energy k and turbulent shear stress normalized with respect to the friction velocity jp for a curved mixing layer... 

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). 

Venneker et al. (2002) used as many as 20 bubble size classes in the bubble size range from 0.25 to some 20 mm. Just like GHOST , their in-house code named DA WN builds upon a liquid-only velocity field obtained with FLUENT, now with an anisotropic Reynolds Stress Model (RSM) for the turbulent momentum transport. To allow for the drastic increase in computational burden associated with using 20 population balance equations, the 3-D FLUENT flow field is averaged azimuthally into a 2-D flow field (Venneker, 1999, used a less elegant simplification )... 

The Reynolds-averaged approach is widely used for engineering calculations, and typically includes models such as Spalart-Allmaras, k-e and its variants, k-co, and the Reynolds stress model (RSM). The Boussinesq hypothesis, which assumes pt to be an isotropic scalar quantity, is used in the Spalart-Allmaras model, the k-s models, and the k-co models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, fit. For the Spalart-Allmaras model, one additional transport equation representing turbulent viscosity is solved. In the case of the k-e and k-co models, two additional transport equations for the turbulence kinetic energy, k, and either the turbulence dissipation rate, s, or the specific dissipation rate, co, are solved, and pt is computed as a function of k and either e or co. Alternatively, in the RSM approach, transport equations can be solved for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (usually for s) is also required. This means that seven additional transport equations must be solved in 3D flows. 

As noted in Chapter 1, the composition PDF description utilizes the concept of turbulent diffusivity (Tt) to model the scalar flux. Thus, it corresponds to closure at the level of the k-e and gradient-diffusion models, and should be used with caution for flows that require closure at the level of the RSM and scalar-flux equation. In general, the velocity, composition PDF codes described in Section 7.4 should be used for flows that require second-order closures. On the other hand, Lagrangian composition codes are well suited for use with an LES description of turbulence. 

Turbulent flows with simple closure models (eddy viscosity, mixing length, k-e) or complex closure models (ASM, RSM, RNG) for the Reynolds stresses... 

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