Models closure

Eddy Viscosity Models. A large number of closure models are based on the Boussinesq concept of eddy viscosity ... 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

More advanced models, for example the algebraic stress model (ASM) and the Reynolds stress model (RSM), are not based on the eddy-viscosity concept and can thus account for anisotropic turbulence thereby giving still better predictions of flows. In addition to the transport equations, however, the algebraic equations for the Reynolds stress tensor also have to be solved. These models are therefore computationally far more complex than simple closure models (Kuipers and van Swaaij, 1997). 

Chen, C. P., Studies in two-phase turbulence closure modeling, Ph.D. Thesis, Michigan State University, USA (1985). 

Particular attention is to be paid to closure models exploiting various types of PDFs such as beta, presumed, or full PDFs (e.g., Baldyga, 1994 Fox, 1996, 2003 Ranade, 2002). While PDFs have successfully been exploited for describing chemical reactions in turbulent flames, tubular reactors (Baldyga and Henczka, 1997), and a Taylor-Couette reactor (Marchisio and Barresi, 2003), they have never been used successfully in stirred reactors so far. 

Baldyga, I., and Henczka, M., Turbulent mixing and parallel chemical reactions in a pipe application of a closure model, Recents Progres en Genie des Procedes 11, 341-348 (1997). Bermingham, S. K., Kramer, H. I. M., and Van Rosmalen, G. M. Comp. Chem. Eng. 22, 355-362... 

Aerosol production and transport over the oceans are of interest in studies concerning cloud physics, air pollution, atmospheric optics, and air-sea interactions. However, the contribution of sea spray droplets to the transfer of moisture and latent heat from the sea to the atmosphere is not well known. In an effort to investigate these phenomena, Edson et al.[12l used an interactive Eulerian-Lagrangian approach to simulate the generation, turbulent transport and evaporation of droplets. The k-e turbulence closure model was incorporated in the Eulerian-Lagrangian model to accurately simulate... 

A general overview of models for turbulent transport is presented in Chapter 4. The goal of this chapter is to familiarize the reader with the various closure models available in the literature. Because detailed treatments of this material are readily available in other texts (e.g., Pope 2000), the emphasis of Chapter 4 is on presenting the various models using notation that is consistent with the remainder of the book. However, despite its relative brevity, the importance of the material in Chapter 4 should not be underestimated. Indeed, all of the reacting-flow models presented in subsequent chapters depend on accurate predictions of the turbulent flow field. With this caveat in mind, readers conversant with turbulent transport models of non-reacting scalars may wish to proceed directly to Chapter 5. 

In Section 4.2, the LES composition PDF was introduced to describe the effect of residual composition fluctuations on the chemical source term. As noted there, the LES composition PDF is a conditional PDF for the composition vector given that the filtered velocity and filtered compositions are equal to U and 0, respectively. The LES composition PDF is denoted by U, 0 x, /), and a closure model is required to describe it. 

In transported PDF methods (Pope 2000), the closure model for A, V, ip) will be a known function26 ofV. Thus, (U,Aj) will be closed and will depend on the moments of U and their spatial derivatives.27 Moreover, Reynolds-stress models derived from the PDF transport equation are guaranteed to be realizable (Pope 1994b), and the corresponding consistent scalar flux model can easily be found. We shall return to this subject after looking at typical conditional acceleration and conditional diffusion models. 

A closure model for homogeneous chemical reactions. Chemical Engineering Science 49, 1985-2003. 

Dutta, A. and J. M. Tarbell (1989). Closure models for turbulent reacting flows. AIChE Journal 35, 2013-2027. 

Hanjalic, K. (1994). Advanced turbulence closure models A view of current status and future prospects. International Journal of Heat and Fluid Flow 15, 178-203. 

Mell, W. E., V. Nilsen, G. Kosaly, and J. J. Riley (1994). Investigation of closure models for nonpremixed turbulent reacting flows. Physics of Fluids 6, 1331-1356. 

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... 

Carl et al. (1973) suggested an expression for X that closely fits a second-order closure model calculation of Zeman and Lumley (1976) ... 

Mellor, G. L., and Yamada, T. (1974). A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci. 31, 1791-1806. 

Wyngaard, J. C., and Cote, O. R. (1974). The evolution of a convective planetary boundary layer—a higher order closure model study. Boundary Layer Meteorol. 7, 289-308. 

The model given above is called the k- -kp model, which can be used for dilute, non-swirling, nonbuoyant gas-solid flows. For strongly anisotropic gas-solid flows, the unified second-order moment closure model, which is an extension of the second-order moment closure model for single-phase flows [Zhou, 1993], may be used. 

The Reynolds-averaged reaction rate Rav(c) could be evaluated by using simple closure models of Bourne and Toor (1977), Brodkey and Lewalle (1985), Li and Toor (1986), Dutta and Tarbell (1989), etc. It should also be pointed out that the spatial averaging presented here is independent of the methodology using which Rilv(c) and D v are evaluated, or in other words, spatial averaging follows time averaging. 

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