Statistical turbulence model

Second, due to the difficulty of accessing multiphase flows with laser-based flow diagnostics, there is very little experimental data available for validating multiphase turbulence models to the same degree as done in single-phase turbulent flows. For example, thanks to detailed experimental measurements of turbulence statistics, there are many cases for which the single-phase k- model is known to yield poor predictions. Nevertheless, in many CFD codes a multiphase k-e model is used to supply multiphase turbulence statistics that cannot be measured experimentally. Thus, even if a particular multiphase turbulent flow could be adequately described using an effective viscosity, in most cases it is impossible to know whether the multiphase turbulence model predicts reasonable values for... 

The transported PDF equation contains more information than an RANS turbulence model, and can be used to derive the latter. We give two example derivations U) and (uuT below, but the same procedure can be carried out to find any one-point statistic of the velocity and/or composition fields.25... 

This means that larger scale motions can be explicitly resolved and deterministically forecasted. The smaller scale motions, namely turbulence, are not explicitly resolved. Rather, the effects of those sub-grid scales on the larger scales are approximated by turbulence models. These smaller size motions are said to be parameterized by sub-grid scale stochastic (statistical) approximations or modes. The referred experimental data analyzes of the flow in the atmospheric boundary layer determine the basis for the large eddy simulation (LES) approach developed by meteorologists like Deardorff [27] [29] [30]. Large-Eddy Simulation (LES) is thus a relatively new approach to the calculation of turbulent flows. 

In this section the heat and mass transport coefficients for turbulent boundary layers are examined. In this case the model derivation is based on the governing Reynolds averaged equations. In these equations statistical covariances appear which involve fluctuating velocities, temperatures and concentrations. The nature of these terms is not known a priori and their effects must by estimated by semi-empirical turbulence modeling. The resulting parameterizations allow us to express the unknown turbulent fluctuations in terms of the mean flow field variables. It is emphasized that the Reynolds equations are not actually solved, merely semi-empirical relations are derived for the wall fluxes through the inner boundary layer. 

Shih T-H, Liou WW, Shabbir A, Zhu J (1995) A New k-e Eddy Viscosity Model for High Reynolds Number Turbulent Elows. Comp Eluids 24(3) 227-238 Smith LM, Reynolds WC (1992) On the Yakhot-Orszag renormalization group method for deriving turbulence statistics and models. Phys Fluids A 4(2) 364-390... 

In engineering computations, the turbulent transport of properties is usually treated in a statistical manner, where computations are concerned with the mean velocities, temperatures, and/or concentrations. This statistical approach, however, masks many of the actual physical processes in the dynamic flow field, which must be recovered by the modeling at some level of the turbulence statistics. This modeling was originally guided by the results from experiments, but currently this guidance can rely on simulations as well. 

The evaluation of these statistical second moments is the goal of turbulence models. These models fall into two categories. First are models in which the turbulent fluxes are expressed in the same functional form as their laminar counterparts, but in which the molecular properties of viscosity, thermal conductivity, and diffusion coefficient are supplemented by corresponding eddy viscosities, conductivities, and diffusivities. The primary distinction is the recognition that the eddy coefficients are properties of the turbulent flow field, not the... 

The terms of the form (m/m/) are called the Reynolds stresses. The RANS equations do not consist of a closed set of equations (there are more unknowns than equations), so if the RANS equations are to be solved, the Reynolds stress terms must be modeled somehow. Typically, this modeling is based on experimental measurements. The application of models developed for macroscale flows to turbulent microchannel flows is dependent on the Reynolds stresses being similar for both cases. Recent experimental evidence suggests a strong similarity between turbulence statistics measured in turbulent microchannel flows and turbulence statistics measured in turbulent pipe and channel flows. Thus, the evidence suggests that turbulent models and codes developed to study macroscale turbulent pipe and channel flow should be applicable to the study of turbulent microchaimel flows. 

As pointed out earlier in Chapter 3 and will be covered in more detail later on in this chapter, the flow in a rotary kiln is typically gas-solid turbulent flow with chemical reactions, mainly combustion. The building blocks behind the user-defined functions (UDF) in commercial CFD codes applied to rotary kiln combustion modeling consist of "renormalization group" (RNG) k-s turbulent model for gas phase and, in the case of pulverized combustion particles, the statistical (stochastic) trajectory model for homogeneous volatile and heterogeneous solid-phase char combustion. The underlying equations are discussed in the next section. 

The model species, total mass, momentum, and energy continuity equations are similar to those presented in Section 13.7 on fluidized bed reactors. Constant values of the gas and liquid phase densities, viscosities, and diffusivities were assumed, as well as constant values of the interphase mass transfer coefficient and the reaction rate coefficient. The interphase momentum transfer was modelled in terms of the Eotvos number as in Clift et al. [1978]. The Reynolds-Averaged Navier-Stokes approach was taken and a standard Computational Fluid Dynamics solver was used. In the continuous liquid phase, turbulence, that is, fluctuations in the flow field at the micro-scale, was accounted for using a standard single phase k-e model (see Chapter 12). Its applicability has been considered in detail by Sokolichin and Eigenberger [1999]. No turbulence model was used for the dispersed gas phase. Meso-scale fluctuations around the statistically stationary state occur and were explicitly calculated. This requires a transient simulation and sufficiently fine spatial and temporal grids. 

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