RANS turbulence models

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). 

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. 

The presentation in this section has been intentionally kept short, as our primary objective is to present the standard form of each model so that the reader can refer to them in later chapters. Nevertheless, it is extremely important for the reader to realize that the quality of a reacting-flow simulation will in no small part depend on the performance of the turbulence model. The latter will depend on a number of issues15 that are outside the scope of this text, but that should not be neglected when applying RANS models to complex flows. 

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We will revisit this topic in Section III when discussing CFD models for mixing-sensitive reactions. Note that while the discussion above applies to RANS turbulence models, the method can be extended to LES by integrating over the SGS wavenumbers (i.e., starting at kc). 

The experienced reader will recognize these CFD models as die so-called RANS turbulence models. 

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... 

At this point, the next step is to decompose the velocity into its mean and fluctuating components, and to substitute the result into the left-hand side of (6.42). In doing so, the triple-correlation term (UiUjUk) will appear. Note that if the joint velocity PDF were known (i.e., by solving (6.19)), then the triple-correlation term could be computed exactly. This is not the case for the RANS turbulence models discussed in Section 4.4 where a model is required to close the triple-correlation term. 

While some of these disadvantages can be overcome by devising improved algorithms, the problem of level of description of the RANS turbulence model remains as the principal shortcoming of composition PDF code. One thus has the option of resorting to an LES description of the flow combined with a composition PDF code, or a less-expensive second-order RANS model using a velocity, composition PDF code. 

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. 

The Smagorinsky Model (cf. Ref. [51]) is an algebraic model in the same spirit as the Prandtl mixing length model discussed in section RANS Turbulence Modeling. In the Smagorinsky model, the SGS stresses are assumed to be proportional to the rate of strain, that is, = VtS, and the kinematic eddy viscosity is determined from the expression... 

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