Modeling the PDF

Modeling the data reveals much more information that straight model independent analysis. The most popular approach for real-space modeling is to use 

Parameters in the structural model, and other experiment-dependent parameters, are allowed to vary until a best-fit of the PDF calculated from the model and the data derived PDF is obtained, using a least-squares approach. The sample dependent parameters thus derived include the unit cell parameters (unit cell lengths and angles), atomic positions in the unit cell expressed in fractional coordinates, anisotropic thermal ellipsoids for each atom and the average atomic occupancy of each site. 

PDFfit was originally designed to study disorder and short-range order in crystalline materials with significant disorder such as nanoporous bulk materials. It has also found extensive use in studying more heavily disordered 

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For premixed flows, the mixture fraction is not applicable. Nonetheless, the methods in this section can still be employed to model the PDF of the reaction-progress variable. 

Chatwin, P. C. 2000. Some Remarks on Modelling the PDF of the Concentration of a Dispersing Scalar in Turbulence, submitted to European Journal of Applied Mathematics. 

A statistical model may depend on a convenient probability distribution function (pdf) for statistically indistinguishable data generation from the actual records of the same phenomena. These models can be categorized into parametric and non-parametric types, where in the former case the pdf s parameters play role, such as the mean and variance in a normal distribution, or the coefficients for the various exponents of the independent variable. However, in case of a nonparamet-ric model the pdf parameters do not enter directly into the model construction but they are only loosely implied by assumptions. In statistics there can be mental (descriptive qualities or physical conceptual in character) event models. 

Determine the spatial extension (correlation length) of the local structure, by modeling the PDF over various sections, for instance, 0-10 A, 10-20 A,... 

Confidence levels on the reliability estimates from the SSI model can be determined and are useful when the PDFs for stress and strength are based on only small amounts of data or where critical reliability projects are undertaken. However, approaches to determine these confidence levels only strictly apply when stress and strength are characterized by the Normal distribution. Detailed examples can be found in Kececioglu (1972) and Sundararajan and Witt (1995). 

As a result, VEGA creates a PDF file that contains all the information about the prediction, including the final assessment of the prediction, the list of the six most similar compounds found in the training and test set of the model, the list of all Applicability Domain indices and a reasoning on SAs with a brief explanation of their meaning. 

In order to implement the PDF equations into a LES context, a filtered version of the PDF equation is required, usually denoted as filtered density function (FDF). Although the LES filtering operation implies that SGS modeling has to be taken into account in order to capture micromixing effects, the reaction term remains closed in the FDF formulation. Van Vliet et al. (2001) showed that the sensitivity to the Damkohler number of the yield of competitive parallel reactions in isotropic homogeneous turbulence is qualitatively well predicted by FDF/LES. They applied the method for calculating the selectivity for a set of competing reactions in a tubular reactor at Re = 4,000. 

The NDF is very similar to the PDFs introduced in the previous section to describe turbulent reacting flows. However, the reader should not confuse them and must keep in mind that they are introduced for very different reasons. The NDF is in fact an extension of the finite-dimensional composition vector laminar flow where the PDFs are not needed, the NDF still introduces an extra dimension (1) to the problem description. The choice of the state variables in the CFD model used to solve the PBE will depend on how the internal coordinate is discretized. Roughly speaking (see Ramkrishna (2000) for a more complete discussion), there are two approaches that can be employed ... 

As seen above, the mean chemical source term is intimately related to the PDF of the concentration fluctuations. In non-premixed flows, the rate of decay of the concentration fluctuations is controlled by the scalar dissipation rate. Thus, a critical part of any model for chemical reacting flows is a description of how molecular diffusion works to damp out... 

In one-point models for turbulent mixing, extensive use of conditional statistics is made when developing simplified models. For example, in the PDF transport equation for /++ x, r), the expected value of the velocity fluctuations conditioned on the scalars appears and is defined by... 

In the LEM, turbulence is modeled by a random rearrangement process that compresses the scalar field locally to simulate the reduction in length scales that results from turbulent mixing. For example, with the triplet map, defined schematically in Fig. 4.2, a random length scale / is selected at a random point in the computational domain, and the scalar field is then compressed by a factor of three.14 The PDF for /,... 

As for the models derived from the PDF transport equation, nearly all widely used models for Ilf can be expressed as... 

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. 

Figure 5.1. Closures for the chemical source term can be understood in terms of their relationship to the joint composition PDF. The simplest methods attempt to represent the joint PDF by its (lower-order) moments. At the next level, the joint PDF is expressed in terms of the product of the conditional joint PDF and the mixture-fraction PDF. The conditional joint PDF can then be approximated by invoking the fast-chemistry or flamelet limits, by modeling the conditional means of the compositions, or by assuming a functional form for the PDF. Similarly, it is also possible to assume a functional form for the joint composition PDF. The best method to employ depends strongly on the functional form of the chemical source term and its characteristic time scales.

The PDF of an inert scalar is unchanged by the first two steps, but approaches the well mixed condition during step (3).108 The overall rate of mixing will be determined by the slowest step in the process. In general, this will be step (1). Note also that, except in the linear-eddy model (Kerstein 1988), interactions between Lagrangian fluid particles are not accounted for in step (1). This limits the applicability of most mechanistic models to cases where a small volume of fluid is mixed into a much larger volume (i.e., where interactions between fluid particles will be minimal). 

Given the strong assumptions needed to model the joint PDF, the use of more refined models would not be justified. 

In a multi-environment micromixing model, the presumed composition PDF has the following form ... 

Note finally that, for any given value of the mixture fraction (i.e., f f), the multienvironment presumed PDF model discussed in this section will predict a unique value of 4>. In this sense, the multi-environment presumed PDF model provides a simple description of the conditional means (0 f) at Ve discrete values of f. An obvious extension of the method would thus be to develop a multi-environment conditional PDF to model the conditional joint composition PDF / (-i/d x, / ). We look at models based on this idea below. 

Ad hoc extensions may be possible for this case by fixing the compositions in one environment at the stoichiometric point, and modeling the probability. On the other hand, by making Ne large, the results will approach those found using transported PDF methods. 

Multi-environment presumed PDF models can be developed for the LES composition PDF using either the unconditional, (5.341), or conditional, (5.396), form. However, in order to simplify the discussion, here we will use the unconditional form to illustrate the steps needed to develop the model. The LES composition PDF can be modeled by163... 

As discussed in Section 4.2, the conditional mean compositions will, in general, depend on the filter so that 4> V, 4>) = 4> need not be true. However, if the equality does not hold, it is then necessary to model the difference. Given the simplicity of the multi-environment presumed PDF, such a complication does not seem warranted. 

The procedure followed above can be used to develop a multi-environment conditional LES model starting from (5.396). In this case, all terms in (5.399) will be conditioned on the filtered velocity and filtered compositions,166 in addition to the residual mixture-fraction vector = - . In the case of a one-component mixture fraction, the latter can be modeled by a presumed beta PDF with mean f and variance (f,2>. LES transport equations must then be added to solve for the mixture-fraction mean and variance. Despite this added complication, all model terms carry over from the original model. The only remaining difficulty is to extend (5.399) to cover inhomogeneous flows.167 As with the conditional-moment closure discussed in Section 5.8 (see (5.316) on p. 215), this extension will be non-trivial, and thus is not attempted here. 

There is no information on the instantaneous scalar dissipation rate and its coupling to the turbulence field. A transported PDF micromixing model is required to determine the effect of molecular diffusion on both the shape of the PDF and the rate of scalar-variance decay. 

In contrast to moment closures, the models used to close the conditional fluxes typically involve random processes. The choice of the models will directly affect the evolution of the shape of the PDF, and thus indirectly affect the moments of the PDF. For example, once closures have been selected, all one-point statistics involving U and 0 can be computed by deriving moment transport equations starting from the transported PDF equation. Thus, in Section 6.4, we will look at the relationship between (6.19) and RANS transport equations. However, we will first consider the composition PDF transport equation. 

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