Turbulence, modeling approaches

Compared with Equation 4.6, Equation 4.12 contains the term -pUi Uj, the so-called Reynolds stress, which represents the effect of turbulence and must be modeled by the CFD code. Limited computational resources restrict the direct simulation of these fluctuations, at least for the moment. Therefore the transport equations are commonly modified to account for the averaged fluctuating velocity components. Three commonly applied turbulence modeling approaches have been used in the CFD model of spray drying system, i.e., k-Q model (Launder and Spalding 1972, 1974), RNG k-e model (Yakhot and Orszag 1986), and a Reynolds stress model (RSM) (Launder et al. 1975). 

CFD simulations at high Reynolds numbers for technical applications are nowadays mainly based on solutions of the Reynolds averaged Navier-Stokes (RANS) equations. The main reason are that they are simple to apply and computationally more efficient than other turbulence modelling approaches such as LES.It is known, however, that in many flow problems the condition of a turbulent equilibrium is not satisfied, i.e., when strong pressure gradients or flow separation occurs, which reduces the prediction accuracy of the results obtained by one-and two-equation turbulence models used to close the RANS equations [13,15]. 

In comparison with Bakker and Van den Akker (1994b) and Venneker et al. (2002), Khopkar et al. (2005) applied a more sophisticated two-fluid approach including a standard k-e turbulence model. Using the incorrect snapshot approach due to Ranade (2002), their simulation results (for gas flow numbers being 4 times higher than those of Bakker and Van den Akker, 1994b) still exhibit major discrepancies with respect to experimental data. One of the... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

In an effort to improve the description of the Reynolds stresses in the rapid distortion turbulence (RDT) limit, the velocity PDF description has been extended to include directional information in the form of a random wave vector by Van Slooten and Pope (1997). The added directional information results in a transported PDF model that corresponds to the directional spectrum of the velocity field in wavenumber space. The model thus represents a bridge between Reynolds-stress models and more detailed spectral turbulence models. Due to the exact representation of spatial transport terms in the PDF formulation, the extension to inhomogeneous flows is straightforward (Van Slooten et al. 1998), and maintains the exact solution in the RDT limit. The model has yet to be extensively tested in complex flows (see Van Slooten and Pope 1999) however, it has the potential to improve greatly the turbulence description for high-shear flows. More details on this modeling approach can be found in Pope (2000). 

While the form of this term is the same as the viscous-dissipation term in the conditional acceleration, the modeling approach is very different. Indeed, while the velocity field in a homogeneous turbulent flow is well described by a multi-variate Gaussian process, the scalar fields are very often bounded and, hence, non-Gaussian. Moreover, joint scalar... 

O. Physical Model Approach for Turbulent Mass Transfer near a Liquid-Fluid Interface... 

The Eulerian continuum approach is basically an extension of the mathematical formulation of the fluid dynamics for a single phase to a multiphase. However, since neither the fluid phase nor the particle phase is actually continuous throughout the system at any moment, ways to construct a continuum of each phase have to be established. The transport properties of each pseudocontinuous phase, or the turbulence models of each phase in the case of turbulent gas-solid flows, need to be determined. In addition, the phase interactions must be expressed in continuous forms. 

With respect to the hydrochemical structure, one can distinguish the southwestern part, which finds itself under the influence of the Bosphorus and represents an area of intensive redox processes in a multilayered transition zone. Here, the chemical conditions are extremely instable due to the temporal and spacial variations in the supply of the Bosphorus waters. In other regions of the Black Sea, the hydrochemical structure is mainly formed and maintained by a combination of biogeochemical and hydrophysical processes such as advection, turbulence, sedimentation, etc. and can be explained with ID-model approach. This leads to the formation of a chemotropic structure, where all features of the chemical parameters distribution are closely correlated with the water density. 

The simplest turbulence model is based on the assumption that relations between e and h (or Prj) and the mean flow variables can be obtained from measurements in relatively simple flow situations and that these relations are then more universally applicable. Such an approach has, however, not met with a great deal of success except in the analysis of free boundary flows, e.g., in jets and wakes. 

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