Turbulence closures

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

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... 

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. 

Hanjalic, K., S. Jakirlic, and I. Hadzic (1997). Expanding the limits of equilibrium second-moment turbulence closures. Fluid Dynamics Research 20, 25 41. 

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... 

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

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 well-known Princeton model with a vertical -coordinate, a curvilinear horizontal grid adapted to the coastline, a turbulent closure of the order of 2.5 was used for the studies of the BSGC in [58]. Eighteen levels were specified over the vertical and the horizontal spacing was about 10 km. Similarly to [48], various combinations of the surface boundary conditions were specified. The model started with the wintertime climatic temperature analysis salinity fields [11] and three years later reached a quasi-stationary regime in the upper 200-m layer. 

The four equations governing two-dimensional turbulent flow are, therefore, Eqs. (2.95), (2.106), and (2.116). These contain, beside the four mean flow variables u, v, p, and T, additional terms which depend on the turbulence held. In order to solve this set of equations, therefore, information concerning these turbulence terms must be available and the difficulties associated with the solution of turbulent flow problems arise basically from the difficulty of analytically predicting the values of these terms. The set of equations effectively contains more variables than the number of equations. This is termed the turbulence closure problem . In order to bring about closure of this set of equations, extra equations must be generated, these extra equations constituting a turbulence model . 

Discuss the meaning of the following terms (i) The turbulence closure problem, (ii) A turbulence model, (iii) A boundary layer, (iv) Similar flow fields. 

Transport (advection and diffusion) of tracers (both passive and reactive) is performed on-line at each meteorological time-step using WAF scheme for advection and a true (second order) diffusion, with diffusion coefficient carefully estimated from experiments (Tampieri and Maurizi 2007). Vertical diffusion is performed using ID diffusion equation with a diffusion coefficient estimated by means of an k-l turbulence closure scheme. Dry deposition is computed through the resistance-analogy scheme and is provided as a boundary condition to the vertical diffusion equation. Furthermore, vertical redistribution of tracers due to moist convection is parameterized consistently with the Kain-Frisch scheme used in the meteorological part for moist convection. Transport of chemical species is performed in mass units while gas chemistry is computed in ppm. 

Chambers, T.L. and Wilcox, D.C. (1977), Critical examination of two-equation turbulence closure models for boundary layers, AIAA J., 15(6), 821. 

Launder, B.E., Reece, G.J. andRodi, W. (1975), Progress in the development of Reynolds stress turbulence closure, J. Fluid Meek, 68, 537. 

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as partial differential equations (PDEs) along with some method of turbulence closure and appropriate initial and boundary conditions. Such models have become more common with the steady increase in computing power and sophistication of numerical algorithms. However, there are many potential problems that must be addressed. In the verification process, the PDEs being solved must adequately represent the physics of the dispersion process especially for processes such as ground-to-cloud heat transfer, phase changes for condensed phases, and chemical reactions. Also, turbulence closure methods (and associated boundary and initial conditions) must be appropriate for the dis-... 

Fully computational methods (FCM), using a variety of turbulence modelling methods, have been extensively applied to computing flows around single buildings in turbulent boundary layers. Using turbulence closure methods many models have predicted mean... 

The more usual approach to speeding up FCM methods for multi-body flow / dispersion calculations such as this is to use the continuum approach. This requires some estimates of turbulence parameters such as mixing lengths or eddy diffusivities for the canopy (on the neighbourhood scale), but these characteristics cannot be deduced from turbulence closure models, although they may be estimated from detailed calculations of flow/dispersion around a few typical obstacles - not a straightforward or accurate process (see Moulinec et al., 2003 [436]). 

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. 

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