Turbulence model table

As mentioned before in Eq. (3), the most common source of SGS phenomena is turbulence due to the Reynolds number of the flow. It is thus important to understand what the principal length and time scales in turbulent flow are, and how they depend on Reynolds number. In a CFD code, a turbulence model will provide the local values of the turbulent kinetic energy k and the turbulent dissipation rate s. These quantities, combined with the kinematic viscosity of the fluid v, define the length and time scales given in Table I. Moreover, they define the local turbulent Reynolds number ReL also given in the table. 

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). 

This conclusion is supported by iwo other comparisons. If the hump in the retention-time curve, which begins at r=1.2 p. is ignored as an anomaly, the time during which measured tracer concentration exceeded one half of the peak concentration was 0.8S/p, which is slightly less than the value expected. This indicates that (he actual flow regime was closer to plug flow than was predicted The lack of turbulence also is indicated by the value of EUd) achieved al peak concentration this was about 10% higher than was predicted by either flow model (Table 2). 

These dimensionless groups also appear in empirical correlations of the turbulent flow region. Although even in the approximate Eq. (9) of Table 6.7, group He appears to affect the friction factor, empirical correlations such as Figure 6.5(b) and the data analysis of Example 6.10 indicate that the friction factor is determined by the Reynolds number alone, in every case by an equation of the form, / = 16/Rc, but with Re defined differently for each model. Table 6.7 collects several relations for laminar flows of fluids. 

For a turbulence model to be useful in a general-purpose CFD code, it must be simple, accurate, economical to run, and have a wide range of applicability. Table 10-1 gives the most common turbulence models. The classical models use the Reynolds equations and form the basis of turbulence calculations in currently available commercial CFD codes. Farge eddy simulations are turbulence models where the time-dependent flow equations are solved for the mean flow and the largest eddies and where the effects of the smallest eddies are modeled. 

Table 8.2 shows typical levels of physical viscosity, turbulent viscosity i/t and artificial viscosity i/a reached in a combustion chamber for a standard regime. All viscosities are scaled by the physical viscosity. Note that viscosity affects all scales and not only the small scales. For example, acoustic waves are very strongly dissipated in a RANS code because the turbulent viscosity acts on them too. This is a collateral effect of turbulence models formulated using turbulent viscosities but it implies that such methods cannot be used for the present objectives. 

As is evident from inspection of Table III turbulence modeling of multiphase flow systems requires major attention in the near future. Also the development of closure laws for phenomena taking place in the vicinity of interfaces such as coalescence, breakup, and accumulation of impurities should be considered in more detail. Once these requirements have been met, in principle, it would be possible to predict a.o. flow regime transition and the spatial distribution of the phases with confidence, which is of utmost importance to the chemical engineer dealing with the design of (novel) multiphase reactors. 

A new term is introduced, the so-called Reynolds stresses m-m). The overbar denotes a time average. This term is the correlation between the turbulent velocity fluctuations and uj, and it describes the transport of momentum in the mean flow due to turbulence. This term is difficult to model, and over the years a variety of turbulence models have been developed. Turbulence models are necessary for calculating time-averaged flow fields directly, without first having to calculate a fully time-dependent flow field and then doing time averaging. The use of turbulence models is therefore much more computationally efficient. A detailed discussion is beyond the scope of this entry, but it is important to note that not all turbulence models are equally suited for all types of flow. Table 1 summarizes the most common turbulence models and their properties. 

Annular. When the vapor velocity is high enough (j > 1.5), gravitational effects can be neglected, and the condensate collects as a thin annular film around the inside of the tube walls, with no stratification. A significant portion of most condensers operate in this flow regime. There are numerous predictive models described in the literature for annular flow. Laminar flow models predict heat transfer coefficients that are too low, and turbulent models must be used. The most commonly used models are listed in Table 14.1. All models have a form for the local Nusselt number... 

Table 19.3 k-s turbulence model constants. (C3 and Cj are for the compressible form)... 

In order to close the additional Reynolds (turbulent) stresses, several different eddy viscosity-based turbulence models, in which the additional turbulent stresses are related to the mean velocity gradient as shown in Table 6.11, are used to account for the turbulence in three-phase systems. Generally, the standard k-e turbulence model is solved only for the continuous phase or for mixture phase or for each phase. In the literature reports. 

Table 5-3 Results of the MRF Impeller Model with Several Turbulence Models as Compared to Experiment...

Turbulent Thermal Diffusivity Model Table 2.1 Model constants of Eq. (2.10) by different aufhcn ... 

The values used for the empirical parameters in the kc-Sc turbulence model used for the continuous gas phase flow are given in Table 4.9,... 

To determine how the scalar time scale defined in Eq. (15) is related to the turbulence integral time scale given in Table I, we can introduce a normalized model scalar energy spectrum (Fox, 2003) as follows ... 

The dominant transport process from water is volatilization. Based on mathematical models developed by the EPA, the half-life for M-hexane in bodies of water with any degree of turbulent mixing (e.g., rivers) would be less than 3 hours. For standing bodies of water (e.g., small ponds), a half-life no longer than one week (6.8 days) is estimated (ASTER 1995 EPA 1987a). Based on the log octanol/water partition coefficient (i.e., log[Kow]) and the estimated log sorption coefficient (i.e., log[Koc]) (see Table 3-2), ii-hexane is not expected to become concentrated in biota (Swann et al. 1983). A calculated bioconcentration factor (BCF) of 453 for a fathead minnow (ASTER 1995) further suggests a low potential for -hcxanc to bioconcentrate or bioaccumulate in trophic food chains. 

The relationship between the CRE approach and the FM approach to modeling turbulent reacting flows is summarized in Table 1.1. Despite the obvious and significant differences... 

Table 1.1. Relationship between the CRE andFM approaches for modeling the important physical processes present in turbulent reacting flows. 

For convenience, the turbulence statistics used in engineering calculations of inhomogeneous, high-Reynolds-number turbulent flows are summarized in Table 2.4 along with the unclosed terms that appear in their transport equations. Models for the unclosed terms are discussed in Chapter 4. 

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