Turbulence modelling, efficiency

The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k-e model, the RSM involves five more differential equations). The number of equations further depends on the character of the simulation whether it is 3-D, 21/2-D, or just 2-D (see below, under The domain and the grid ). In the case of two-phase flow simulations, the use of two-fluid models implies doubling the number of NS equations required for single-phase flow. All this may urge the development of more efficient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. 

M 39] [P 37] The Reynolds-stress model describes best the experimental findings out of three turbulent models investigated (see Figure 1.105) [41]. Then, the model was used for predictions of the mixing efficiency as determined by an azo-type parallel reaction. It was found that the wall thickness has no major influence, whereas the channel depth, as expected, has an influence, affecting the shearing. 

A new term is introduced, the so-called Reynolds stresses m-m). The overbar denotes a time average. This term is the correlation between the turbulent velocity fluctuations and uj, and it describes the transport of momentum in the mean flow due to turbulence. This term is difficult to model, and over the years a variety of turbulence models have been developed. Turbulence models are necessary for calculating time-averaged flow fields directly, without first having to calculate a fully time-dependent flow field and then doing time averaging. The use of turbulence models is therefore much more computationally efficient. A detailed discussion is beyond the scope of this entry, but it is important to note that not all turbulence models are equally suited for all types of flow. Table 1 summarizes the most common turbulence models and their properties. 

The effect of drag reducers on the turbulence is modelled with computational fluid dynamics (CFD) by using a two-layer turbulence model. In the laminar buffer layer, the one-equation model of Hassid and Poreh (1975) is used to describe the enhanced dissipation of turbulence caused by drag reducers. The standard k-e model is applied in the fully turbulent regions. The flow conditions necessary to elongate the polymer, the drag reduction efficiency of polymers of different apparent molar masses and their degradation kinetics have been measured. This data has been used in the model development. 

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]. 

Figure 5-38 Flow pattern at the surface of a vessel equipped with a high efficiency impeller calculated using an LES turbulence model.

A more precise approach is to compute gas convection directly by solving the Navier-Stokes equation including an appropriate turbulence model. However, this increases the numerical efforts considerably and one needs efficient numerical methods using a combination of block-structured and unstructured meshes. 

Another design method uses capture efficiency. There are fewer models for capture efficiency available and none that have been validated over a wide range of conditions. Conroy and Ellenbecker - developed a semi-empirical capture efficiency for flanged slot hoods and point and area sources of contaminant. The point source model uses potential flow theory to describe the flow field in front of a flanged elliptical opening and an empirical factor to describe the turbulent diffusion of contaminant around streamlines. 

In order to achieve that an environmental fate model is successfully applied in a screening level risk assessment and ultimately incorporated into the decisionmaking tools, the model should have computational efficiency and modest data input. Moreover, the model should incorporate all relevant compartments and all sources of contamination and should consider the most important mechanisms of fate and transport. Although spatial models describe the environment more accurately, such models are difficult to apply because they require a large amount of input data (e.g., detailed terrain parameters, meteorological data, turbulence characteristics and other related parameters). Therefore, MCMs are more practical, especially for long-term environmental impact evaluation, because of their modest data requirements and relatively simple yet comprehensive model structure. In addition, MCMs are also widely used for the comparative risk assessment of new and existing chemicals [28-33]. 

Fig. 2 (left panel) shows that by adopting the a calibration by Ludwig et al. (1999), we obtain evolutionary tracks very similar to those obtained by means of the Full Spectrum Turbulence (FST) model (Canuto, Goldman Mazzitelli 1996), which is also known to be a high efficiency model. 

When solid body rotation is assumed in the CE (model B), the degree of differential rotation at its base is too low to trigger efficient shear-induced turbulence between the outer part of the hydrogen burning shell (HBS) and the CE (solid lines in Fig. la). On the contrary, in our model C the differential rotation... 

An attempt has been made by Tsouris and Tavlarides[5611 to improve previous models for breakup and coalescence of droplets in turbulent dispersions based on existing frameworks and recent advances. In both the breakup and coalescence models, two-step mecha-nisms were considered. A droplet breakup function was introduced as a product of droplet-eddy collision frequency and breakup efficiency that reflect the energetics of turbulent liquid-liquid dispersions. Similarly, a coalescencefunction was defined as a product of droplet-droplet collision frequency and coalescence efficiency. The existing coalescence efficiency model was modified to account for the effects of film drainage on droplets with partially mobile interfaces. A probability density function for secondary droplets was also proposed on the basis of the energy requirements for the formation of secondary droplets. These models eliminated several inconsistencies in previous studies, and are applicable to dense dispersions. 

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