Standard Reynolds Stress Model

It is noted that in recent papers several extended turbulence models, i.e., the standard k-e [50], RNG k-e [97, 98, 99, 82, 66], realizable k-e [81], Chen-Kim k-e [11], optimized Chen-Kim k-e [44], standard k-uj [96], k-uj shear-stress transport (SST) [56, 57, 58] and the standard Reynolds stress models, have been proposed and validated. However, little or no significant improvements have been achieved considering the predictivity of the turbulence models, although each of them may have minor advantages and disadvantages. A few... 

The model equations of standard Reynolds stress model are ... 

The standard Reynolds stress model has anisotropic character and thus can give better result than the k — 8 model in predicting anisotropic flow, although more equations need be solved. Note that the model constants may be adjusted for different flow problems. 

Similar to the standard Reynolds stress model, the algebraic model involves twelve variables and twelve model equations. The model is closed and solvable. 

Reynolds stress model, in which the unknown pw-w, is calculated directly by the modeled Eq. (1.23a). The advantage of this model is anisotropy and more rigorous, while its weakness is the need of much computer work. It is called standard Reynolds stress model. For reducing the computer load, a simplified model, called Algebraic Reynolds stress model, Eq. (1.24) is used instead of Eq. (1.23a). The accuracy of simplified model is comparable to the standard Reynolds stress model. 

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. 

At least three approaches have been proposed to solve for the mean pressure field that avoid the noise problem. The first approach is to extract the mean pressure field from a simultaneous consistent39 Reynolds-stress model solved using a standard CFD solver.40 While this approach does alleviate the noise problem, it is intellectually unsatisfying since it leads to a redundancy in the velocity model.41 The second approach seeks to overcome the noise problem by computing the so-called particle-pressure field in an equivalent, but superior, manner (Delarue and Pope 1997). Moreover, this approach leads to a truly... 

The case of an impeller downward velocity of 5ms was further investigated to examine the influence of turbulence models and discretization schemes. The influence of discretization schemes on predicted results (with the standard k-s model) is shown in Fig. 7.7. It can be seen that with a sufficiently fine grid, the influence of the discretization scheme is not significant. Additional simulations reveal that for the coarser grid there is a significant difference in the predicted results of different discretization schemes. The difference diminishes as the number of computational cells increases. The influence of the turbulence model employed on predicted results is shown in Fig. 7.8. It can be seen that the predictions by standard and RNG versions of k-s models are almost the same. The predictions of the Reynolds stress model are, however, significantly different from these two models. This illustrates the importance of appropriate selection of turbulence model and the... 

Consequently, although the second-order closure models is considered a standard model in most commercial CFD codes, the Reynolds stress model is usually not considered worthwhile for complex reactor simulations. Actually, for dynamic simulations the interpretation problems, mentioned earlier in this paragraph, have shifted the attention towards the VLES simulations to be described shortly. In this book the second-order closure models are thus not considered in further details, the interested reader is referred to standard textbooks on turbulence modeling for CFD applications (e.g., [186] [121]). 

The Reynolds averaging approach has thus been found to represent a tradeoff between accuracy and computational costs. Using the Reynolds averaging concept turbulence is interpreted as a waveform and described by the time averaged equations of motion and a turbulence closure. Two-equation turbulence models like the standard k-e and k-cu models are common but full Reynolds stress models have also been applied in rare cases. 

Haque et al [35] simulated turbulent flows with a free-surface in unbaffled agitated vessels using three turbulence models (i.e., the Reynolds stress, standard k-e and k-ix) SST models) as implemented in CFX. In this case it appears that the SST model did perform better than the standard k-e model, whereas some features of the flow structure and the mean velocity profiles were better predicted by the Reynolds stress model. 

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... 

Zakrzewska and Jaworski [100] performed singlephaseCFD simulations of turbulent jacket heat transfer in a Rushton turbine stirred vessel using the eight turbulence models mentioned above as implemented in FLUENT. In all simulations the boundary flow at the vessel wall was described by the standard logarithmic wall functions. The predicted values of the local heat transfer coefficient were compared with measured values. In these simulations the standard k-s, the optimized Chen-KimA - and the k-Lo SST model results were in fair agreement with experimental data, whereas the realizable k-s, RNG k-e and the Reynolds stress model were not recommended. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

Standard k-s The most widely used model, it is robust, economical, and time tested. The Reynolds stresses are not calculated directly, but are modeled in a simplified way by adding a so-called turbulent viscosity to the molecular viscosity. Its main advantages are a rapid, stable calculation, and reasonable results for many flows, especially those with a high Reynolds number. It is not recommended for highly swirling flows, round jets, or flows with strong flow separation... 

The averaging process creates a new set of variables, the so-called Reynolds stresses, which are dependent on the averages of products of the velocity fluctuations UjUj (which for i = j simply represent the standard deviations of the velocity components). This creates a closure problem, which is one of the fundamental issues that has to be addressed in the modeling of turbulent flows. Importantly, Equation 3.2.12 also indicates that the Reynolds stress terms, which in line with Taylor s fundamental result should be related to the dispersion parameters, are coupled to the gradients of the mean flow velocity. 

The standard two-equation k-e model has been used for almost all of the simulations referred to in this chapter because it is the most tested and reliable turbulence model available. Although it will not give the amount of information that a mean Reynolds stress or an algebraic stress model will give, it requires an order of magnitude less CPU time and gives predictions of the mean velocities that are of comparable accuracy to the higher order models. 

Two-Equation Models of Turbulence. On application of Reynolds time averaging, six new unknowns (the Reynolds stresses) appear in the momentum equations. There are now more unknowns than eqnations, so the system of equations is no longer closed. This is the closure problem of tmbulence. Physical flow models for the Reynolds stresses are needed to close the eqnations. Many logical closme schemes have been proposed and have met with some snccess for certain classes of flows, but there is no standard, fnlly validated approach to the modeling of Reynolds stresses. 

Realizable k-e Model. The realizable k-e model (Shih et al., 1995) is a fairly recent addition to the family of two-equation models. It differs from the standard k-e model in two ways. First, the turbulent viscosity is computed in a different manner, making use of eq. (5-16) but using a variable for the quantity C, . This is motivated by the fact that in the limit of highly strained flow, some of the normal Reynolds stresses, u , can become negative in the k-e formulation, which is unphysical, or unrealizable. The variable form of the constant C, is a function of the local strain rate and rotation of the fluid and is designed to prevent unphysical values of the normal stresses from developing. 

The governing equations of a standard DEM presented below include the local average fluid model of Anderson-Jackson [3] type, where the viscous effect is lumped on to the vicinity of particle surface and the Reynolds stress term is neglected, and the Newton s equation of motion for each particle of the same size, which is assumed to be soft spheres so that multiple collisions. 

The PGT model represents an extension of the FT models in that the gas turbulence is taken into account by including the Reynolds stress tensor in the momentum equation for the gas phase. The turbulence model used for the gas phase is similar to the standard single phase k-e turbulence model presented in Sect. 1.3.5, although additional generation and dissipation terms may be added to consider the presence of particles. In the PGT model the drift velocity is neglected. 

In order to close the additional Reynolds (turbulent) stresses, several different eddy viscosity-based turbulence models, in which the additional turbulent stresses are related to the mean velocity gradient as shown in Table 6.11, are used to account for the turbulence in three-phase systems. Generally, the standard k-e turbulence model is solved only for the continuous phase or for mixture phase or for each phase. In the literature reports. 

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