Turbulence, Reynolds stress models

Reynolds Stress Models. Eddy viscosity is a useful concept from a computational perspective, but it has questionable physical basis. Models employing eddy viscosity assume that the turbulence is isotropic, ie, u u = u u = and u[ u = u u = u[ = 0. Another limitation is that the... 

The flow pattern is ealeulated from eonservation equations for mass and mometum, in eombination with the Algebraie Stress Model (ASM) for the turbulent Reynolds stresses, using the Fluent V3.03 solver. These equations ean be found in numerous textbooks and will not be reiterated here. Onee the flow pattern is known, the mixing and transport of ehemieal speeies ean be ealeulated from the following model equation ... 

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. 

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). 

Although Eq. (6-18) can be used to eliminate the stress components from the general microscopic equations of motion, a solution for the turbulent flow field still cannot be obtained unless some information about the spatial dependence and structure of the eddy velocities or turbulent (Reynolds) stresses is known. A classical (simplified) model for the turbulent stresses, attributed to Prandtl, is outlined in the following subsection. 

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. 

The last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

However, if a Reynolds-stress model is used to describe the turbulence, a modified gradient-diffusion model can be employed ... 

The GLM thus contains 12 model parameters which can be chosen to agree with any realizable Reynolds-stress model (Pope 1994b). Pope and co-workers have made detailed comparisons between the GLM and turbulent-flow data. In general, the agreement is good for flows where the corresponding Reynolds-stress model performs adequately. 

Despite the ability of the GLM to reproduce any realizable Reynolds-stress model, Pope (2002b) has shown that it is not consistent with DNS data for homogeneous turbulent shear flow. In order to overcome this problem, and to incorporate the Reynolds-number effects observed in DNS, a stochastic model for the acceleration can be formulated (Pope 2002a Pope 2003). However, it remains to be seen how well such models will perform for more complex inhomogeneous flows. In particular, further research is needed to determine the functional forms of the coefficient matrices in both homogeneous and inhomogeneous turbulent flows. 

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). 

Dreeben, T. D. and S. B. Pope (1997a). Probability density function and Reynolds-stress modeling of near-wall turbulent flows. Physics of Fluids 9, 154—163. 

Taulbee, D. B., F. Mashayek, and C. Barre. 1999. Simulation and Reynolds stress modeling of particle-laden turbulent shear flows. Int. J. Heat Fluid Flow 20(4) 368-73. 

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],... 

Computational fluid dynamics were used to describe the flow which undergoes a fast transition from laminar (at the fluid outlets) to turbulent (in the large mixing chamber) [41]. Using the commercial tool FLUENT, the following different turbulence models were applied a ke model, an RNC-ki model and a Reynolds-stress model. For the last model, each stream is solved by a separate equation for the two first models, two-equation models are applied. To have the simulations at... 

M 39] [P 37] The Reynolds-stress model describes best the experimental findings out of three turbulent models investigated (see Figure 1.105) [41]. Then, the model was used for predictions of the mixing efficiency as determined by an azo-type parallel reaction. It was found that the wall thickness has no major influence, whereas the channel depth, as expected, has an influence, affecting the shearing. 

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... 

Hanjalic, K. and Launder, B.E. (1972), A Reynolds stress model of turbulence and its application to thin shear flows, J. Fluid Mech., 52, 609. 

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]). 

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