Turbulence model RANS equations

These two transport equations for k and e form an inherent part of any k i model of RANS-simulations. As the result of closing the turbulence modeling such that no further unknown variables and equations are introduced, the e-equation does contain some terms that are still the result of modeling, albeit at the very small scales (e.g., Rodi, 1984). 

In Section 2.2, the Reynolds-averaged Navier-Stokes (RANS) equations were derived. The resulting transport equations and unclosed terms are summarized in Table 2.4. In this section, the most widely used closures are reviewed. However, due to the large number of models that have been proposed, no attempt at completeness will be made. The reader interested in further background information and an in-depth discussion of the advantages and limitations of RANS turbulence models can consult any number of textbooks and review papers devoted to the topic. In this section, we will follow most closely the presentation by Pope (2000). 

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... 

Unlike Lagrangian composition codes that use two-equation turbulence models, closure at the level of second-order RANS turbulence models is achieved. In particular, the scalar fluxes are treated in a consistent manner with respect to the turbulence model, and the effect of chemical reactions on the scalar fluxes is treated exactly. 

In turbulent reactive flows, the chemical species and temperature fluctuate in time and space. As a result, any variable can be decomposed in its mean and fluctuation. In Reynolds-averaged Navier-Stokes (RANS) simulations, only the means of the variables are computed. Therefore, a method to obtain a turbulent database (containing the means of species, temperature, etc.) from the laminar data is needed. In this work, the mean variables are calculated by PDF-averaging their laminar values with an assumed shape PDF function. For details the reader is referred to Refs. [16, 17]. In the combustion model, transport equations for the mean and variances of the mixture fraction and the progress variable and the mean mass fraction of NO are solved. More details about this turbulent implementation of the flamelet combustion model can also be found in Ref. [20],... 

To simulate turbulent flows, Reynolds-averaged Navier-Stokes (RANS) equations form the basis for most codes. Several turbulence models are usually provided. A new turbulence model may also usually be incorporated via user-defined routines. Recently, many of the commercial CFD codes have announced the inclusion of large eddy simulation (LES) capabilities. Considering the importance of rotating equipment used in reactor engineering applications, the ability to handle multiple reference frames or sliding meshes is important. Most leading commercial CFD codes provide... 

In general CFD models show a good applicability for risk assessments in urban areas however, their results can differ depending on turbulent closure models and other assumptions. Many CFD models, based on the RANS equations, use the standard k-e turbulence models (originally developed for hydro-dynamical engineering problems), which are violated in complex flow in street canyons and have to be improved and further verified. LES models show substantially better correspondence with measurement data in urban areas and have good perspectives in future, but they are more expensive computationally, and therefore, their usage is limited. 

By far the most widely employed models for turbulent reactive flows in stirred tanks are based on the Reynolds averaged Navier Stokes (RANS) equation. This is a moment equation containing quantities that are averaged over the whole wave spectra, as explained in sect 1.2.7. 

Due to the importance of turbulence in spray systems, this topic is treated in some detail. It includes a summary of time averaging and spatial filtering, followed by a description of RANS and LES turbulence modeling. The RANS model that is presented is the k—s turbulence model, and the LES SGS models that are outlined include the Smagorinsky model and the one-equation subgrid scale (SGS) model. 

Averaging or filtering of a system of conservation equations leads to an identical set of equations for the filtered variables, plus additional, unknown expressions which involve averaged fluctuation terms. This constitutes the notorious closure problem, namely, there are more unknowns than equations, which leaves the system of equations underdetermined. In order to resolve this closure problem, additional relations are required that describe the new unknown fluctuation variables and thereby close the system. These relations are called turbulence models. If the averaging process is done with a time filter then one obtains RANS equations, whereas a spatial filtering leads to LES. 

The averaging process introduces the additional unknown fluctuation terms, u and T, for which no additional information is available. Consequently, there are more unknowns than equations, which is the reason why these expressions need to be modeled. The modeling of the Reynolds stress tensor is the focus of RANS-based turbulence models. 

The second-order correlation of the fluctuations a b is not known and does not appear in the Navier-Stokes equations. Additional equations need to be provided, therefore giving rise to the closure problem. The closures are provided for an area called turbulence modeling for RANS (Reynolds-averaged Navier-Stokes) and LES (large eddy simulation) methodologies. 

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. 

The flow in the gas channels and in the porous gas diffusion electrodes is described by the equations for the conservation of momentum and conservation of mass in the gas phase. The solution of these equations results in the velocity and pressure fields in the cell. The Navier-Stokes equations are mostly used for the gas channels while Darcy s law may be used for the gas flow in the GDL, the microporous layer (MPL), and the catalyst layer [147]. Darcy s law describes the flow where the pressure gradient is the major driving force and where it is mostly influenced by the frictional resistance within the pores [145]. Alternatively, the Brinkman equations can be used to compute the fluid velocity and pressure field in porous media. It extends the Darcy law to describe the momentum transport by viscous shear, similar to the Navier-Stokes equations. The velocity and pressure fields are continuous across the interface of the channels and the porous domains. In the presence of a liquid phase in the pore electrolyte, two-phase flow models may be used to account for the interaction between the gas phase and the liquid phase in the pores. When calculating the fluid flow through the inlet and outlet feeders of a large fuel cell stack, the Reynolds-averaged Navier-Stokes (RANS), k-o), or k-e turbulence model equations should be used due to the presence of turbulence. 

CFD simulations at high Reynolds numbers for technical applications are nowadays mainly based on solutions of the Reynolds averaged Navier-Stokes (RANS) equations. The main reason are that they are simple to apply and computationally more efficient than other turbulence modelling approaches such as LES.It is known, however, that in many flow problems the condition of a turbulent equilibrium is not satisfied, i.e., when strong pressure gradients or flow separation occurs, which reduces the prediction accuracy of the results obtained by one-and two-equation turbulence models used to close the RANS equations [13,15]. 

The RANS simulations use the one-equation turbulence model of Fares and Schroder [7] to close the averaged equations. 

The new terms involving u u are called the Reynolds stresses. The overbar indicates that these terms represent time-averaged values. Reynolds stresses contribute new unknowns to the RANS equations and need to be related to the other variables. This is done through various models, collectively known as turbulence models. 

For turbulent expiratory conditions, avoiding the intensive computational efforts involved with a three-dimensional Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), a Reynolds-Averaged Navier Stokes (RANS) equations coupled to a Shear Stress Transport (SST) fc- y turbulent model is used to model the fluid. The governing equations are essentially similar to (1) and (2) above, but with the inclusion of Reynolds stress... 

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