Turbulence Modeling Analogues

In this subsection an overview of the various formulations of the k — e turbulence models applied to multiphase flows is given. 

A non-isotropic version of the eddy viscosity hypothesis is sometimes used in situations where such effects are essential and the particular flow in question is expected to be far from isotropic [93]. That is, when the turbulent kinetic energy is calculated, the Reynolds stresses can be approximated by  

A k — Model for Bubbly Flows Bubble Induced Turbulence 

To enable simulations of two-phase bubbly flows [67] [65] the single phase k — e model has been extended by including a semi-empirical production term to take into account the additional turbulence production induced by the bubbles motion relative to the liquid (i.e., based on the idea of [128] [129]). In this approach it was assumed that the internal flow inside the dispersed phase (gas bubbles) does not affect the liquid phase turbulence. The shear work performed on the liquid by a single bubble, representing the additional turbulence production due to the bubble, was thus assumed to be equal to the product of the drag force and the relative velocity. 

In vector notation the two-phase k — e model equations for incompressible flows (time averaged) were formulated as follows  

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Early models used a value for that remained constant throughout the day. However, measurements show that the deposition velocity increases during the day as surface heating increases atmospheric turbulence and hence diffusion, and plant stomatal activity increases (50—52). More recent models take this variation of into account. In one approach, the first step is to estimate the upper limit for in terms of the transport processes alone. This value is then modified to account for surface interaction, because the earth s surface is not a perfect sink for all pollutants. This method has led to what is referred to as the resistance model (52,53) that represents as the analogue of an electrical conductance... 

The velocity distributions normal and parallel to the tie rods, respectively, were obtained from potential flow analogue studies of a two-dimensional model of the lower plenum. A wake of highly turbulent, but essentially stagnant fluid, was assumed to exist, protruding downwards into the plenum from the lower edge of the core barrel. The extent of the wake was adjusted to give consistency with the experimentally observed flow non-uniformity at the lower core plate. 

Other driving force units and corresponding coefficients can be used in analogous definitions. Equation (2.4-2) explicitly allows for the bulk flow contribution to the molar flux relative to stationary coordinates. The coefficient or its analogues is thought to exhibit a somewhat less complex relationship to composition, flow conditions, and geometry. Further discussion of the relationships among those coefficients is provided in Section 2.4-1 in terms of particular models for turbulent mass transfer. 

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