Reynolds averaged RANS

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). 

Thus, the reactor will be perfectly mixed if and only if = at every spatial location in the reactor. As noted earlier, unless we conduct a DNS, we will not compute the instantaneous mixture fraction in the CFD simulation. Instead, if we use a RANS model, we will compute the ensemble- or Reynolds-average mixture fraction, denoted by ( ). Thus, the first state variable needed to describe macromixing in this system is ( ). If the system is perfectly macromixed, ( ) = < at every point in the reactor. The second state variable will be used to describe the degree of local micromixing, and is the mixture-fraction variance (maximum value of the variance at any point in the reactor is ( )(1 — ( )), and varies from zero in the feed streams to a maximum of 1/4 when ( ) = 1/2. 

Unsteady RANS (URANS), 11 781. See also Reynolds- averaged Navier-Stokes (RANS)... 

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). 

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],... 

The three main numerical approaches used in turbulence combustion modeling are Reynolds averaged Navier Stokes (RANS) where all turbulent scales are modeled, direct numerical simulations (DNS) where all scales are resolved and large eddy simulations (LES) where larger scales are explicitly computed whereas the effects of smaller ones are modeled ... 

Reynolds-averaged Navier-Stokes Equations (RANS)... 

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... 

RANS Reynolds-averaged Navier-Stokes equations... 

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. 

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. 

Three different theoretical approaches have been established describing turbulent flows in general, as outlined in sect 1.3. These methods are the direct numerical simulations (DNS), large eddy simulations (LES), and the Reynolds average Navier-Stokes (RANS) approach. 

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. 

When dealing with turbulent flows all the relevant dimensionless numbers are evaluated with the available quantities. For example, in DNS, the fluid and particle instantaneous velocities will be employed, whereas in large-eddy simulation (LES) or in Reynolds-average Navier-Stokes-equations (RANS) simulations the filtered or Reynolds-average values will be used. 

RANS Reynolds-average Navier-Stokes equation... 

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