Turbulence Models Based on RANS

A large proportion of the models of Reynolds stress use an eddy viscosity hypothesis based on an analogy between molecular and turbulent motions. Accordingly, turbulent eddies are visualized as molecules, colliding and exchanging momentum and obeying laws similar to the kinetic theory of gases. This allows the description of Reynolds stresses  

is referred to as turbulent or eddy viscosity, which, in contrast to molecular viscosity, is not a fluid property but depends on the local state of flow or turbulence. It is assumed to be a scalar and may vary significantly within the flow domain, k is the turbulent kinetic energy (normal turbulent stresses) and can be expressed as 

Substitution of Eq. (3.14) in the Reynolds-averaged momentum conservation equations (Eqs (3.11)) leads to a closed set, provided the turbulent viscosity is known. The form of the Reynolds-averaged momentum equations remain identical to the form of the laminar momentum equations (Chapter 2 and Table 2.2) except that molecular viscosity is replaced by an effective viscosity, 

By analogy with the kinetic theory of gases, turbulent viscosity may be related to the characteristic velocity and length scales of turbulence (uj and Ij respectively)  

The turbulence models then attempt to devise suitable methods/equations to estimate these characteristic length and velocity scales to close the set of equations. 

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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. 

An advanced turbulence modeling based on hybridization of large eddy simulation (LES) and Reynolds-averaged Navier-Stokes equations (RANS) allowing one to use the best of both worlds ... 

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. 

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. 

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. 

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. 

Based on 2-D RANS and 3-D DNS simulations at meso-scale, it was concluded that 3-dimensional transient RANS simulations are probably the most promising approach for accurate and relatively inexpensive modelling at the macro (whole-body) scale. This approach allows the inclusion of various realistic and important phenomena, such as natural convection and chemical breakthrough. The results of this DNS simulation are nsed (i) to validate RANS and T-RANS simulations for the same conditions and (ii) to guide in choosing a turbulence modelling strategy for the RANS and T-RANS simulations of the flow underneath the porous layer. 

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]. 

In RANS-based simulations, the focus is on the average fluid flow as the complete spectrum of turbulent eddies is modeled and remains unresolved. When nevertheless the turbulent motion of the particles is of interest, this can only be estimated by invoking a stochastic tracking method mimicking the instantaneous turbulent velocity fluctuations. Various particle dispersion models are available, such as discrete random walk models (among which the eddy lifetime or eddy interaction model) and continuous random walk models usually based on the Langevin equation (see, e.g.. Decker and... 

Sommerfeld, 2000 Dehbi, 2008). Given the various and numerous assumptions of any turbulence model used for the RANS simulation and the degree of sophistication of the random walk model appHed, a very accurate representation of the fluid—particle interaction force may not be very essential. Hence, very realistic information on particle trajectories cannot be expected on the basis of RANS-based flow fields. 

In real life, the parcels or blobs are also subjected to the turbulent fluctuations not resolved in the simulation. Depending on the type of simulation (DNS, LES, or RANS), the wide range of eddies of the turbulent-fluid-flow field is not necessarily calculated completely. Parcels released in a LES flow field feel both the resolved part of the fluid motion and the unresolved SGS part that, at best, is known in statistical terms only. It is desirable that the forces exerted by the fluid flow on the particles are dominated by the known, resolved part of the flow field. This issue is discussed in greater detail in the next section in the context of tracking real particles. With a RANS simulation, the turbulent velocity fluctuations remaining unresolved completely, the effect of the turbulence on the tracks is to be mimicked by some stochastic model. As a result, particle tracking in a RANS context produces less realistic results than in an LES-based flow field. 

Several papers on Euler—Euler simulations (both RANS based and of the above EELES type) report about the need to include a separate turbulent dispersion term (or force) for reproducing a correct spatial distribution of suspended particle clouds or bubble swarms. Various models for this turbulent dispersion have been suggested, all of them modeling this effect in terms of the gradient of the particle or bubble volume fraction. Sometimes, this additional term is in the continuity equations (Bakker and Van den Akker, 1994 Sardeshpande and Ranade, 2012 Tamburini et al, 2014) with the justification that is due to fluctuations in the volume fractions however, as the volume fraction which anyhow is an averaged variable does not vary... 

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