RANS models Reynolds stresses

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. 

RANS turbulence models are the workhorse of CFD applications for complex flow geometries. Moreover, due to the relatively high cost of LES, this situation is not expected to change in the near future. For turbulent reacting flows, the additional cost of dealing with complex chemistry will extend the life of RANS models even further. For this reason, the chemical-source-term closures discussed in Chapter 5 have all been formulated with RANS turbulence models in mind. The focus of this section will thus be on RANS turbulence models based on the turbulent viscosity hypothesis and on second-order models for the Reynolds stresses. 

RANS, under which the Reynolds-averaged Navier Stokes equations are solved using some type of closure assumption to account for the Reynolds stress terms. RANS provides the values of the mean wind velocity and estimates of the turbulence statistics within the model domain. 

All numerical models incorporate significant assumptions and approximations, and their predictions must always be regarded as estimates. Solution of the RANS equations, for example, requires some form of closure assumption dealing with the Reynolds stress terms. Since the Reynolds stress terms and the mean flow terms are coupled by the equations, inaccuracies in the closure approximations can affect the predicted mean flow fleld. Furthermore, the boundary conditions imposed on the model require the assumption of velocity profiles and momentum transport rates, which may themselves be approximated. Similar approximations are inherent in any of the various techniques used to compute the wind fleld, with further assumptions being present in each of the dispersion models. 

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 basic idea of RANS models is to account for the change in the fluid transport properties by introducing an eddy viscosity, I, also called turbulence viscosity, which relates the Reynolds stress tensor R to the fluid deformation. Such a relationship was first proposed by Boussinesq in the nineteenth century. More formally, this Boussinesq assumption can be written as... 

The terms that do require modeling are the subgrid stresses, C and R. In fact, most SGS models account for the entire SGS stress given in (19.36). Consequently, LES modeling is formally similar to the modeling of the Reynolds stress tensor in the RANS approach and, therefore, analogous methods are used for LES... 

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. 

In the RANS-approach, turbulence or turbulent momentum transport models are required to calculate the Reynolds-stresses. This can be done starting from additional transport equations, the so-called Reynolds-stress models. Alternatively, the Reynolds-stresses can be modeled in terms of the mean values of the variables and the turbulent kinetic energy, the so-called turbulent viscosity based models. In either way, the turbulence dissipation rate has to be calculated also, as it contains essential information on the overall decay time of the velocity fluctuations. In what follows, the more popular models based on the turbulent viscosity are focused on. A detailed description of the Reynolds-stress models is given in Annex 12.5.l.A which can be downloaded from the Wiley web-page. 

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... 

Reynolds stress model (RANS) in which the Reynolds stress transport equation is computed directly by modeling Eq. (1.6) to the form suitable for numerical computation ... 

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