RANS models equation

The scalar statistics used in engineering calculations of high-Reynolds-number turbulent mixing of an inert scalar are summarized in Table 3.2 along with the unclosed terms that appear in their transport equations. In Chapter 4, we will discuss methods for modeling the unclosed terms in the RANS transport equations. 

Thus, solutions to the transported PDF equation will provide more information than is available from second-order RANS models without the problem of closing the chemical source term. 

In contrast to moment closures, the models used to close the conditional fluxes typically involve random processes. The choice of the models will directly affect the evolution of the shape of the PDF, and thus indirectly affect the moments of the PDF. For example, once closures have been selected, all one-point statistics involving U and 0 can be computed by deriving moment transport equations starting from the transported PDF equation. Thus, in Section 6.4, we will look at the relationship between (6.19) and RANS transport equations. However, we will first consider the composition PDF transport equation. 

Time dependence The inherent time dependence of fires sets strong requirements for computational efficiency. In RANS models, the radiation field must be updated within the internal iterations of the time step, but the computational cost can be relaxed by solving RTE only every Mh iteration. In SOFIE, for example, it is typical to use N = 10. In FDS, the time accuracy of the radiation field has been relaxed by solving the FVM equations typically every third time step and only part of the directions at a time. 

Although less well defined than turbulence of the continuous phase, the dispersed, particle phase also experiences turbulence. Turbulence in the solids is usually either treated with a simple RANS model, such as k-e, or ignored. There are other considerations, such as the influence turbulence in one phase has on turbulence in the other phase. These effects are not captured in the momentum transfer terms contained in the time averaged equations and must be separately included. 

Due to the importance of turbulence in spray systems, this topic is treated in some detail. It includes a summary of time averaging and spatial filtering, followed by a description of RANS and LES turbulence modeling. The RANS model that is presented is the k—s turbulence model, and the LES SGS models that are outlined include the Smagorinsky model and the one-equation subgrid scale (SGS) model. 

The k-E model equations, given in Table 19.5, are for a compressible fluid. In comparison with the standard k-E model for incompressible flows, given in (19.34) and (19.35), there are additional terms due to compressibility, involving div u, and due to the multiphase nature of the flow, given by the spray source term IT . The constants used in this model are summarized in Table 19.3. As mentioned above, there is no spray source term IT necessary when LES is used instead of RANS. 

The flow in the gas channels and in the porous gas diffusion electrodes is described by the equations for the conservation of momentum and conservation of mass in the gas phase. The solution of these equations results in the velocity and pressure fields in the cell. The Navier-Stokes equations are mostly used for the gas channels while Darcy s law may be used for the gas flow in the GDL, the microporous layer (MPL), and the catalyst layer [147]. Darcy s law describes the flow where the pressure gradient is the major driving force and where it is mostly influenced by the frictional resistance within the pores [145]. Alternatively, the Brinkman equations can be used to compute the fluid velocity and pressure field in porous media. It extends the Darcy law to describe the momentum transport by viscous shear, similar to the Navier-Stokes equations. The velocity and pressure fields are continuous across the interface of the channels and the porous domains. In the presence of a liquid phase in the pore electrolyte, two-phase flow models may be used to account for the interaction between the gas phase and the liquid phase in the pores. When calculating the fluid flow through the inlet and outlet feeders of a large fuel cell stack, the Reynolds-averaged Navier-Stokes (RANS), k-o), or k-e turbulence model equations should be used due to the presence of turbulence. 

In this particular case, the first (1st) and second (2nd) ran isotherms are virtually coincident indicating that CO adsorption was entirely reversible upon evacuation of the CO equilibrium pressure. For the experimental and samples details vide infra Sect. 1.4. It is here only recalled that the 2nd ran isotherms were performed after the overnight outgassing of the reversible adsorbed phase. The isotherms experimental points were interpolated by the Langmuir model equation (vide infra). 

Figure 4. Fits of lattice strain model to experimental mineral-melt partition coefficients for (a) plagioclase (run 90-6 of Blundy and Wood 1994) and (b) elinopyroxene (ran DC23 of Blundy and Dalton 2000). Different valence cations, entering the large cation site of each mineral, are denoted by different symbols. The curves are non-linear least squares fits of Equation (1) to the data for each valence. Errors bars, when larger than symbol, are 1 s.d. Ionic radii in Vlll-fold coordination are taken from Shannon (1976).

Usually, however, the stresses are modeled with the help of a single turbulent viscosity coefficient that presumes isotropic turbulent transport. In the RANS-approach, a turbulent or eddy viscosity coefficient, vt, covers the momentum transport by the full spectrum of turbulent scales (eddies). Frisch (1995) recollects that as early as 1870 Boussinesq stressed turbulence greatly increases viscosity and proposed an expression for the eddy viscosity. The eventual set of equations runs as... 

These two transport equations for k and e form an inherent part of any k i model of RANS-simulations. As the result of closing the turbulence modeling such that no further unknown variables and equations are introduced, the e-equation does contain some terms that are still the result of modeling, albeit at the very small scales (e.g., Rodi, 1984). 

An alternative approach (e.g., Patterson, 1985 Ranade, 2002) is the Eulerian type of simulation that makes use of a CDR equation—see Eq. (13)—for each of the chemical species involved. While resolution of the turbulent flow down to the Kolmogorov length scale already is far beyond computational capabilities, one certainly has to revert to modeling the species transport in liquid systems in which the Batchelor length scale is smaller than the Kolmogorov length scale by at least one order of magnitude see Eq. (14). Hence, both in RANS simulations and in LES, species concentrations and temperature still fluctuate within a computational cell. Consequently, the description of chemical reactions and the transport of heat and species in a chemical reactor ask for subtle approaches as to the SGS fluctuations. 

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). 

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... 

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. 

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