Turbulence closure model description

A variety of statistical models are available for predictions of multiphase turbulent flows [85]. A large number of the application oriented investigations are based on the Eulerian description utilizing turbulence closures for both the dispersed and the carrier phases. The closure schemes for the carrier phase are mostly limited to Boussinesq type approximations in conjunction with modified forms of the conventional k-e model [87]. The models for the dispersed phase are typically via the Hinze-Tchen algebraic relation [88] which relates the eddy viscosity of the dispersed phase to that of the carrier phase. While the simplicity of this model has promoted its use, its nonuniversality has been widely recognized [88]. 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

The interfacial and turbulence closures suggested in the literature also differ considering the anticipated importance of the bubble size distributions. It thus seemed obvious for many researchers that further progress on the flow pattern description was difficult to obtain without a proper description of the interfacial coupling terms, and especially on the contact area or projected area for the drag forces. The bubble column research thus turned towards the development of a dynamic multi-fluid model that is extended with a population balance module for the bubble size distribution. However, the existing models are still restricted in some way or another due to the large cpu demands required by 3D multi-fluid simulations. 

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

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 framework and the basic equations for the EE description are first recalled in Section 10.1. The LES filtering procedure for these equations is described in Section 10.1 along with the closure assumptions and models for subgrid terms, including turbulent combustion. The application is a sector of a gas turbine burner for which both the steady flow and an ignition sequence using a spark discharge are computed in Sections 10.2 and 10.2... 

In summary, one of the weakest links in modeling reactive systems operated in bubble columns is the fluid dynamic part considering multi-phase turbulence modeling, interfacial closures, and especially the impact and descriptions of bubble size and shape distributions. For reactive systems the estimates of the contact areas and thus the interfacial mass transfer rates are likely to contain large uncertainties. 

Another aspect is the description of the turbulence in the gas phase due to the movement of the particles. Recent efforts aim to close the Euler—Euler models with appropriate closures for the turbulence and turbulence modulation, or describe the interaction among particles to form clusters which... 

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. 

Abstract The computational mass transfer (CMT) aims to find the concentration profile in process equipment, which is the most important basis for evaluating the process efficiency as well as the effectiveness of an existing mass transfer equipment. This chapter is dedicated to the description of the fundamentals and the recently published models of CMT for obtaining simultaneously the concentration, velocity and temperature distributions. The challenge is the closure of the differential species conservation equation for the mass transfer in a turbulent flow. Two models are presented. The first is a two-equation model termed as model,... 

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