K S model

For effective mixing, high Reynolds numbers are required and the turbulence should be dissipated in the bulk of the flow and as little as possible on the walls. The source term for generating turbulent energy is in the k—s model written as... 

In another class of models, pioneered by Elghobashi and Abou-Arab (1983) and Chen (1985), a particle turbulent viscosity, derived by extending the concept of turbulence from the gas phase to the solid phase, has been used. This is the so-called k—s model, where the k corresponds to the granular temperature and s is a dissipation parameter for which another conservation law is required. By coupling with the gas phase k—s turbulence model, Zhou and Huang (1990) developed a k—s model for turbulent gas-particle flows. The k—s models do not... 

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. 

Using the definition for the turbulent viscosity (jit — /An /xmoi), which gives a result similar to the standard k-s model with only a small difference in the modeling constant, the effective viscosity is now defined as a function of k and s in Eq. (16) in algebraic form. 

The differential form of this equation is used in calculating the effective viscosity in the RNG k-s model. This method allows varying the effective viscosity with the effective Reynolds number to accurately extend the model to low-Reynolds-number and near-wall flows. 

The transport equations for the turbulent kinetic energy, k, and the turbulence dissipation, e, in the RNG k-s model are again defined similar to the standard k-s model, now utilizing the effective viscosity defined through the RNG theory. The major difference in the RNG k-s model from the standard k-s model can be found in the e balance where a new source term appears, which is a function of both k and s. The new term in the RNG k- s model makes the turbulence in this model sensitive to the mean rate of strain. The result is a model that responds to the effect of strain and the effect of streamline curvature,... 

The idea involved in the k-s model is to assume that the Reynolds stress tensor can be written as... 

In addition, for highly nonlinear cases, the mesh size needed to avoid spurious solutions may be so small that this approach is not feasible, (ii) The CFD approach also uses averaged models (e.g., k-s model for turbulent flows) with closure schemes that are not always justified and contain adjustable constants. 

The renormalized k-s model of FLUENT is used to compute the Reynolds stress tensors r, and the effective properties for each phase. A k-s turbulence model is also used in [3] and [8]. A less CPU-consuming model based on mixture-lengths is used in [4],... 

The modeled transport equations for z differ mainly in the diffusion and secondary source term. Launder and Spalding (1972) and Chambers and Wilcox (1977) discuss the differences and similarities in more detail. The variable, z = e is generally preferred since it does not require a secondary source, and a simple gradient diffusion hypothesis is fairly good for the diffusion (Launder and Spalding, 1974 Rodi, 1984). The turbulent Prandtl number for s has a reasonable value of 1.3, which fits the experimental data for the spread of various quantities at locations far from the walls, without modification of any constants. Because of these factors, the k-s model of turbulence has been the most extensively studied and used and is recommended as a baseline model for typical internal flows encountered by reactor engineers. 

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 ... 

It must be remembered that since all the assumptions may not be valid for flows of practical interest, the model parameters are not truly universal but are functions of characteristic flow parameters. Several attempts have been made to enhance the applicability of the k-s model by modifying these empirical parameters to suit the specific requirements of different types of flow. One of the weaknesses of the standard... 

In most high Reynolds number flows, the wall function approach gives reasonable results without excessive demands on computational resources. It is especially useful for modeling turbulent flows in complex industrial reactors. This approach is, however, inadequate in situations where low Reynolds number effects are pervasive and the hypotheses underlying the wall functions are not valid. Such situations require the application of a low Reynolds number model to resolve near-wall flows. For the low Reynolds number version of k-s models, the following boundary conditions are used at the walls ... 

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. 

Algebraic Stress Models (ASM) Accounts for anisotropy Combines generality of approach with the economy of the k-s model Good performance for isothermal and buoyant thin shear layers Restricted to flows where convection and diffusion terms are negligible Performs as poorly as k-e in some flows due to problems with s equation Not widely validated... 

Eddy lifetime or the integral time scale of turbulence can be expressed in the framework of the k-s model as... 

Subscript 1 indicates continuous phase and 2 indicates dispersed phase. Cd is a parameter of the standard k-s model (0.09), k is turbulent kinetic energy and si is turbulent energy dissipation rate. The eddy lifetime seen by dispersed phase particles will in general be different from that for continuous phase fluid particles due to the so-called crossing-trajectory effect (Csnady, 1963). This can be expressed in the form ... 

Scalar equations (if any) are then solved using the corrected velocity field (for example, k and s equations when solving the k-s model of turbulence or the enthalpy equation when solving non-isothermal flows). 

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... 

FIGURE 7,5 Predicted flow field with two equation turbulence model (standard k-s model), (a) Vector plot, (b) Contours of stream function (-0.01 [A] to 0.01 [J]), (c) Contours of turbulent kinetic energy (0 [A] to 0.01 [K]). 

Turbulence was modeled using the standard k-s model. All the governing equations were discretized using a QUICK discretization scheme with SUPERBEE limiter function (Fluent User Guide, 1997). The SIMPLE algorithm (Patankar, 1980) was... 

In order to understand the possible mal-distribution, it is essential to make an accurate prediction of flow in the upper region of the reactor, where severe changes in flow directions occur. Typical values of throughput for the RFR under consideration indicate that the flow is turbulent (for the specific case modeled here, feed velocity at the inlet was 40 m s ). The selection of an appropriate turbulence model is, therefore, crucial. Anticipating recirculating flow in the upper region of the reactor with spatial variation of velocity and length scales of turbulence, it will be necessary to use at least a two-equation turbulence model. The standard k-s model of turbulence, which has been tested and found to be useful for a variety of applications, may be used in absence of more specific information. 

RNG k-E A modified version of the ks model, this model yields improved results for swirling flows and flow separation. It is not well suited for round jets and is not as stable as the standard k-s model... 

An analysis [238] of different simulation calculations of flow conditions in baffled vessels with turbine stirrers proposed in the literature resulted in a number of inconsistencies and provoked the question of whether it was permissible to simplify the three dimensional flow by an axially symmetrical approximation. The results obtained for a turbulent 3-D single phase flow, showed that the standard k-s-model did not produce a satisfactory result, due to the strongly pronounced velocity gradients in the turbulence field in the immediate proximity of the stirrer. In particular, the necessary assumption of a momentum sink for the baffles averaged over the circumference of the tank led to misrepresentation of the tangential flow field. 

To close the k-s model we need to eliminate the length scale variable, L, from the model relation (1.403). This has been achieved by relating the length scale to the dissipation rate and k through a semi-empirical relationship ... 

This turbulence model is similar to the standard k-s model, but with altered model parameter values and the effect of swirl on turbulence is included in the RNG mode intending to enhance the accuracy of swirling flow simulations. 

Fig. 8.9. Axial velocity-, gas voidage- and turbulent viscosity profiles as a function of column radius at the axial level z = 2.0 (m) after 80 (s) (steady-state) employing the steady drag and added mass forces. Crosses experimental data [61], continuous line standard k-s model, case (a), dotted hne standard k-s model plus Sato model, case (b), dashed line extended k-s model, case (c). Grid resolution 20x72, time resolution 2 10 " (s). Reprinted with permission from [66]. Copyright 2005 American Chemical Society.

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