Turbulence second-order closure models

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... 

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

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]). 

Ma.s,small, W. J. and Weil, 1. C. (1999). An analytical one-dimensional second-order closure model of turbulence statistics and the Lagrangian time scale within and above plant canopies of arbitrary structure. Botindary-Layer Meteorol. 91, 81-107. 

For practical applications, second-order closure models are required for the third-order diffusion correlations, the pressure-strain correlation and the dissipation rate correlation as described by Launder and Spalding [95] and Wilcox ([185], Sect. 6.3). Launder and Spalding [95] argued that the pressure diffusion terms and the molecular diffusion of turbulent momentum fluxes are smaller than the rest of the terms in the equation. These terms can thus be sufficiently approximated by a gradient... 

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. 

The calculation of the six components of the Reynolds stress tensor, that is, six second-order moments of the micro-PDF, f v,yf), is reduced to the calculation of k and the modeling of the turbulent viscosity pf As seen from (12.5.1-2), is a function of a limited number of second-order moments of the micro-PDF. Turbulent viscosity based closure models for the Reynolds-stresses can be used at relatively low computational effort. In the two-equation model approach, the turbulent viscosity is expressed in terms of the turbulent kinetic energy, k, and the turbulence dissipation rate, s, according to ... 

Donaldson, C. (1975). On the modeling of the scalar correlations necessary to construct a second-order closure description of turbulent reacting flows, in Turbulent Mixing in Nonreactive and Reactive Flows, S. N. B. Murthy, ed.. Plenum Press, New York, pp. 131-162. 

Besides applying the postulation similar to the Boussinesq s (or Pick s law) to solve the Reynolds mass flux — mJc in terms of isotropic turbulent mass diffusivity Dt as described in preceding Sect. 3.2 by c — Sc two-equation model, another model has been recently developed to solve the anisotropic Reynolds mass flux —M-c directly instead of using D, to close the turbulent species mass conservation equation. The Reynolds mass flux model discussed in this section could be known as a result following the turbulence closure postulations for the second-order closure turbulence model in the book of Chen and Jaw [23]. 

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. 

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. 

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. 

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. 

The second-order correlation of the fluctuations a b is not known and does not appear in the Navier-Stokes equations. Additional equations need to be provided, therefore giving rise to the closure problem. The closures are provided for an area called turbulence modeling for RANS (Reynolds-averaged Navier-Stokes) and LES (large eddy simulation) methodologies. 

Wang Q, Squires KD, Simonin O (1998) Large eddy simulation of turbulent gas-solid flows in a vertical channel and evaluation of second-order models. Int J Heat Fluid Flow 19 505-511 Wang Y, Chao Z, Jakobsen HA (2010) A sensitivity study of the two-fluid model closure parameters (/ , e) determining the main gas-soUd flow pattern characteristics. Ind Eng Chem Res 49 3433-3441... 

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