The 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. 

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

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. 

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... 

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... 

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). 

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 ... 

Dash We used the k-s model with correction on compressibility effects. Boundary conditions were more sophisticated than in other approaches. 

In these equations, m, denotes the components of the turbulence-average velocity. Equations [8.39] are summed over indices i and j. The k-s model uses five constants Cju, C, C2, cxk, and cts, whose values were calibrated on the basis of reference experiments. The quantity ... 

It is important to note that the equations of the k-s model are entirely governed by the values of Wnns and it. 

The equations of the k-e model ate coupled and notrlinear, which ittakes their resolution non-trivial. Prove that the two equatiorrs of the k-s model can be combined to establish that ... 

Comment upon the solution obtained when C2 = 2. Why is it necessary that C2 < 2 for the solution of the k-s model to be physically meaningful ... 

These classical models are nowadays embedded in certain numerical codes, and they are most cormnonly used in engineering processes. It is useful to know them, as well as the k-s model suimnarized in Chapter 8. 

RNGk-s A modified version of the k-s 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. 

RSM The full Reynolds stress model provides good predictions for all types of flows, including swirl, separation, and round and planar jets. Because it solves transport equations for the Reynolds stresses directly, longer calculation times are required than for the k-s models. 

In the quasi-single-fluid models, the rising gas-liquid mixture is treated like a homogenous fluid of reduced density, and one set of continuity and momentum equations is solved for the two-phase mixture. The quasi-single-phase modeling technique has been relatively more popular and has been used extensively by Szekely and co-workers [34,42,43] and Guthrie and co-workers [1,2,15,25,44 7]. The k—s model is often used to represent turbulence. In most applications, the void fraction distribution is assumed a priori rather than being solved for, and this limits the predictive capability of these models. 

Due to their robustness and reasonable accuracy, the first-order two-equation models, such as the k-s closure originally proposed by Harlow and Nakayama [62, 63], have become very popular for reactor simulations. In this section the formal derivation of the k-s model equations are given and discussed. A transport equation for the turbulent kinetic energy, or actually the momentum variance, can be derived by multiplying the equation for the fluctuating component v[, (1.395), by 2v[, there-... 

Starting out with fully developed channel flows the quantities of interest (i.e., Vx, k and e) depend only on y, so the k-s model equations (1.407) and (1.410) reduce to ... 

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