Reynolds stresses, modeling

Keywords Computational fluid dynamics (CFD) Reynolds averaging Closure of Reynolds-averaged Navier-Stokes equations Two-equation model Reynolds stress model... 

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... 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

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 ... 

Almost all modern CFD codes have a k - model. Advanced models like algebraic stress models or Reynolds stress model are provided FLUENT, PHOENICS and FLOW3D. Table 10-3 summarizes the capabilities of some widely used commercial CFD codes. Other commercially CFD codes can be readily assessed on the web from hptt//www.cfd-online.com This is largest CFD site on the net that provides various facilities such as a comprehensive link section and discussion forum. 

Turbulence modeling capability (range of models). Eddy viscosity k-1, k-e, and Reynolds stress. k-e and Algebraic stress. Reynolds stress and renormalization group theory (RNG) V. 4.2 k-e. low Reynolds No.. Algebraic stress. Reynolds stress and Reynolds flux. k- Mixing length (user subroutine) and k-e. 

Chen, Q. Prediction of room air motion by Reynolds-stress models. Build. FInviron., vol. 31, pp. 233-244, 1996. 

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. 

Using turbulenee models, this new system of equations ean be elosed. The most widely used turbulenee model is the k-e model, whieh is based on an analogy of viseous and Reynolds stresses. Two additional transport equations for the turbulent kinetie energy k and the turbulent energy dissipation e deseribe the influenee of turbulenee... 

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). 

Tahry, S.E., Application of a Reynolds stress model to engine-like flow calculations. /. of Fluids Engineering, 1985.107(4) 444-450. 

Ekambra etal. [21] compared the results from ID, 2D, and 3D simulations of a bubble column with experimental results. They obtained similar results for holdup and axial velocity, while eddy viscosity, Reynolds stresses, and energy dissipation were very different in the three simulations as shown in Figure 15.7. This example also illustrates the importance of selecting the right variables for model vahdation. A 2D model will yield good results for velocity but will predict all variables based on turbulent characteristics poorly. 

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. 

All these findings of disappointing quantitative agreement with experimental data stem from the inherent drawback of the RANS-approach that there is no clear distinction between the turbulent fluctuations modeled by the Reynolds stresses and (mesoscale) fluctuations. In LES, however, the distinction between resolved and unresolved turbulence is clear and relates to the cell size of the computational grid chosen. 

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 velocities and other solution variables are now represented by Reynolds-averaged values, and the effects of turbulence are represented by the Reynolds stresses, (—pu pTl) that are modeled by the Boussinesq hypothesis ... 

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 reader familiar with turbulence modeling will recognize the covariance matrix as the Reynolds stresses. Thus, for fully developed homogeneous turbulence, knowledge of the mean velocity and the Reynolds stresses completely determines the one-point joint velocity PDF. 

As we shall see in Chapter 4, models for izf have much in common with those used for IZij in the Reynolds stress transport equation. Indeed, as shown using transported PDF methods in Chapter 6, the model for uniquely determines the model for 7zf. 

The form of (4.21) is very similar to that of (2.93), p. 47, for the mean velocity (U) found by Reynolds averaging. However, unlike the Reynolds stresses, the residual stresses depend on how the filter function G is defined (Pope 2000). Perhaps the simplest model for the residual stress tensor was proposed by Smagorinsky (1963) ... 

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