RANS equations

Even nowadays, a DNS of the turbulent flow in, e.g., a lab-scale stirred vessel at a low Reynolds number (Re = 8,000) still takes approximately 3 months on 8 processors and more than 17 GB of memory (Sommerfeld and Decker, 2004). Hence, the turbulent flows in such applications are usually simulated with the help of the Reynolds Averaged Navier- Stokes (RANS) equations (see, e.g., Tennekes and Lumley, 1972) which deliver an averaged representation of the flow only. This may lead, however, to poor results as to small-scale phenomena, since many of the latter are nonlinearly dependent on the flow field (Rielly and Marquis, 2001). 

Now that the state variables have been determined, we can go to steps (ii) and (iii), which involve finding closed CFD transport equations. The derivation of the RANS equations is described in detail in Fox (2003), and will not be repeated here. Instead, we will simply give the CFD transport equations and discuss the closures appearing in the equations. The five transport equations are... 

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). 

By far, the most widely employed models for reactive flow processes are based on Reynolds-averaged Navier Stokes (RANS) equations. As discussed earlier in Chapter 3, Reynolds averaging decomposes the instantaneous value of any variable into a mean and fluctuating component. In addition to the closure equations described in Chapter 3, for reactive processes, closure of the time-averaged scalar field equations requires models for (1) scalar flux, (2) scalar variance, (3) dissipation of scalar variance, and (4) reaction rate. Details of these equations are described in the following section. Broadly, any closure approach can be classified either as a phenomenological, non-PDF (probability density function) or as a PDF-based approach. These are also discussed in detail in the following section. 

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. 

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. 

Reynolds averaged Navier-Stokes (RANS) equation Equation representing the conservation of momentum in a fluid flow, subjected to a temporal or spatial averaging process in line with the approach proposed by Osborne Reynolds. 

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. 

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 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. 

Eq. (25.3) and the subsequent time averaging yield the Navier-Stokes equations for averaged flow variables [Reynolds-averaged Navier-Stokes (RANS) equations] ... 

Turbulent flow is described by conservation equations of continuity and momentum, known as the Reynolds-averaged Navier-Stokes (RANS) equations. Laminar velocity terms in conservation equations are replaced by the steady-state mean components and time-dependent fluctuating components defined by Equation 6.100. 

CFD starts from the Reynolds-Averaged Navier-Stokes (RANS) equations for the calculation of the macro-scale behavior expressed by the macro-scale averaged, also called Reynolds-averaged or mean variables — the... 

As already mentioned in Section 12.3, the unclosed correlation terms that appear in the RANS equations are the Reynolds-stresses... 

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]. 

To reduce the modeling problem to a single steady solution, Reynolds formulated time-averaging rules. Application of these rules yields a time-averaged form of the Navier-Stokes and other equations, known as the Reynolds averaged, or RANS, equations. These equations now relate time-averaged quantities, not instantaneous time-dependent values. For this simplification, we pay a dear price in that there are now more unknowns than 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. 

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