Turbulence model, second-order

Validation and Application. VaUdated CFD examples are emerging (30) as are examples of limitations and misappHcations (31). ReaUsm depends on the adequacy of the physical and chemical representations, the scale of resolution for the appHcation, numerical accuracy of the solution algorithms, and skills appHed in execution. Data are available on performance characteristics of industrial furnaces and gas turbines systems operating with turbulent diffusion flames have been studied for simple two-dimensional geometries and selected conditions (32). Turbulent diffusion flames are produced when fuel and air are injected separately into the reactor. Second-order and infinitely fast reactions coupled with mixing have been analyzed with the k—Z model to describe the macromixing process. 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

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 last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

In summary, in the equilibrium-chemistry limit, the computational problem associated with turbulent reacting flows is greatly simplified by employing the presumed mixture-fraction PDF method. Indeed, because the chemical source term usually leads to a stiff system of ODEs (see (5.151)) that are solved off-line, the equilibrium-chemistry limit significantly reduces the computational load needed to solve a turbulent-reacting-flow problem. In a CFD code, a second-order transport model for inert scalars such as those discussed in Chapter 3 is utilized to find ( ) and and the equifibrium com-... 

As noted in Chapter 1, the composition PDF description utilizes the concept of turbulent diffusivity (Tt) to model the scalar flux. Thus, it corresponds to closure at the level of the k-e and gradient-diffusion models, and should be used with caution for flows that require closure at the level of the RSM and scalar-flux equation. In general, the velocity, composition PDF codes described in Section 7.4 should be used for flows that require second-order closures. On the other hand, Lagrangian composition codes are well suited for use with an LES description of turbulence. 

Relative to velocity, composition PDF codes, the turbulence and scalar transport models have a limited range of applicability. This can be partially overcome by using an LES description of the turbulence. However, consistent closure at the level of second-order RANS models requires the use of a velocity, composition PDF code. 

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. 

In an attempt to circumvent some of these problems, considerable effort has been expended to develop so-called second moment turbulent closure models in which the governing equations are closed by including terms parameterizing various turbulent correlations (see, for example, Lewellen et al., 1974 Wyngaard and Cote, 1974 Lumley and Khajeh-Nouri, 1974 Mellor and Yamada, 1974 Yamada and Mellor, 1975 Zeman and Lumley, 1976, 1979 Zeman and Tennekes, 1977 Freeman, 1977 Yamada, 1977 Manton, 1979 Binkowski, 1979). While second-order closure models are conceptually very appealing, their use in atmospheric... 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

It can be seen from the above consideration that the theory of EPR flows based on the introduction of the distributed force and sources of substances agrees well with experimental data, at least qualitatively. At the same time, the notable scattering can motivate looking for a refined theory, especially as concerned to the algebraic turbulence model (3.131). Second-order turbulence models that are expressed in terms of the differential equations for vT and the associated quantities should certainly fit better,... 

It can thus be thought that the intensive turbulence within EPRs, i.e. canopies, reveals some features that are very distinguishing from the common unobstructed turbulence. Such kind of the turbulence attracted an increased attention of researchers in last years, [81, 155, 186, 187, 305, 318, 410, 462, 500, 522], Despite the simplified first-order turbulence closures (algebraic models) or second-order ones (with differential equations for vr) turned useful and lead to some plausible results in practical areas, many its phenomena remains unexplained. Further information about basic turbulence laws is provided in Chapters 2, 4 to 9 along with further practical applications. 

Katul, G., and Chang, W.H. (1999) Principal length scales in second-order closure models for canopy turbulence, J. Appl. Meteor.38, 1631-1643. 

Umlauf, L., Biuchard, H, 2005, Second-order turbulence closure models for geophysical boundary layers. A review of recent work. Continental Shelf Research, 25, 795-827. 

Sykes, R.L and Gabruk, R.S., 1997. A second-order closure model for the effect of averaging time on turbulent plume dispersion, J. Appl. Meteorol., 36, pp. 1038-1045. 

The next category of turbulence closures, i.e., impl3ung to be more accurate than the very simple algebraic models, is a hierarchy of turbulent models based on the transport equation for the fluctuating momentum field. These are the first-order closure models, i.e., those that require parameterizations for the second moments and the second-order closure models, i.e., those that... 

Consequently, although the second-order closure models is considered a standard model in most commercial CFD codes, the Reynolds stress model is usually not considered worthwhile for complex reactor simulations. Actually, for dynamic simulations the interpretation problems, mentioned earlier in this paragraph, have shifted the attention towards the VLES simulations to be described shortly. In this book the second-order closure models are thus not considered in further details, the interested reader is referred to standard textbooks on turbulence modeling for CFD applications (e.g., [186] [121]). 

Wang Q, Squires KD, Simonin O (1998) Large eddy simulation of turbulent gas-solid flows in a vertical chaimel and evaluation of second-order models. Int J Heat and Fluid Flow 19 505-511... 

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