Probability density function model

Erratum Application of the velocity-dissipation probability density function model to inhomogeneous turbulent flows [Phys. Fluids A 3, 1947 (1991)]. Physics of Fluids A Fluid Dynamics 4, 1088. 

Pozorski, J., and J. P. Minier. 1999. Probability density function modeling of dispersed two-phase turbulent flow. Phys. Rev. E 59 855-63. 

Zaichik, L., Oesterle, B. Aupchenkov, V. 2004 On the probability density function model for the transport of particles in anisotropic turbulent flow. Physics of Eluids 16 (6), 1956-1964. 

Probability Density Function model (PDF) Discrete Ordinate model (DO) Euler-Lagrangian model... 

Figure 4 Sample spatial restraint m Modeller. A restraint on a given C -C , distance, d, is expressed as a conditional probability density function that depends on two other equivalent distances (d = 17.0 and d" = 23.5) p(dld, d"). The restraint (continuous line) is obtained by least-squares fitting a sum of two Gaussian functions to the histogram, which in turn is derived from many triple alignments of protein structures. In practice, more complicated restraints are used that depend on additional information such as similarity between the proteins, solvent accessibility, and distance from a gap m the alignment.

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

The thermally produced noise voltage X(t) appearing across the terminals of a hot resistor is often modeled by assuming that the probability density function for X(t) is gaussian,... 

Luo and Domfeld [110] introduced a fitting parameter H , a d5mamical" hardness value of the wafer surface to show the chemical effect and mechanical effect on the interface in their model. It reflects the influences of chemicals on the mechanical material removal. It is found that the nonlinear down pressure dependence of material removal rate is related to a probability density function of the abrasive size and the elastic deformation of the pad. 

A model must be introduced to simulate fast chemical reactions, for example, flamelet, or turbulent mixer model (TMM), presumed mapping. Rodney Eox describes many proposed models in his book [23]. Many of these use a probability density function to describe the concentration variations. One model that gives reasonably good results for a wide range of non-premixed reactions is the TMM model by Baldyga and Bourne [24]. In this model, the variance of the concentration fluctuations is separated into three scales corresponding to large, intermediate, and small turbulent eddies. 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

Principles of mathematical modelling 2 Probability density function 112 Process control examples 505-524 Product inhibition 643, 649 Production rate in mass balance 27 Profit function 108 Proportional... 

If the mathematical model of the process under consideration is adequate, it is very reasonable to assume that the measured responses from the i,h experiment are normally distributed. In particular the joint probability density function conditional on the value of the parameters (k and ,) is of the form,... 

The multimedia model present in the 2 FUN tool was developed based on an extensive comparison and evaluation of some of the previously discussed multimedia models, such as CalTOX, Simplebox, XtraFOOD, etc. The multimedia model comprises several environmental modules, i.e. air, fresh water, soil/ground water, several crops and animal (cow and milk). It is used to simulate chemical distribution in the environmental modules, taking into account the manifold links between them. The PBPK models were developed to simulate the body burden of toxic chemicals throughout the entire human lifespan, integrating the evolution of the physiology and anatomy from childhood to advanced age. That model is based on a detailed description of the body anatomy and includes a substantial number of tissue compartments to enable detailed analysis of toxicokinetics for diverse chemicals that induce multiple effects in different target tissues. The key input parameters used in both models were given in the form of probability density function (PDF) to allow for the exhaustive probabilistic analysis and sensitivity analysis in terms of simulation outcomes [71]. 

The key input parameters used in the 2 FUN model were given in the form of probability density function (PDF) to allow the exhaustive probabilistic analysis and sensitivity analysis in terms of simulation outcomes. 

Davis, J.M. (1997a). Justification of probability density function for resolution in statistical models of overlap. Chromatographia 44, 81. 

Differential pressure measurements were made between several vertical elevations within the bed. The probability density function of the cold model and combustor gave very close agreement (Fig. 35). The solid fraction profiles were obtained from the vertical pressure profile with a hydrostatic assumption. The cold model solid fraction profile showed very close agreement with data taken from pressure taps in two different locations within the combustor (Fig. 36). The solid fraction shows a... 

The importance of chemical-reaction kinetics and the interaction of the latter with transport phenomena is the central theme of the contribution of Fox from Iowa State University. The chapter combines the clarity of a tutorial with the presentation of very recent results. Starting from simple chemistry and singlephase flow the reader is lead towards complex chemistry and two-phase flow. The issue of SGS modeling discussed already in Chapter 2 is now discussed with respect to the concentration fields. A detailed presentation of the joint Probability Density Function (PDF) method is given. The latter allows to account for the interaction between chemistry and physics. Results on impinging jet reactors are shown. When dealing with particulate systems a particle size distribution (PSD) and corresponding population balance equations are intro-... 

Using the field model described in section 1, detection probabilities are to be computed for each grid point to find the breach probability. The optimal decision rule that maximizes the detection probability subject to a maximum allowable false alarm rate a is given by the Neyman-Pearson formulation [20]. Two hypotheses that represent the presence and absence of a target are set up. The Neyman-Pearson (NP) detector computes the likelihood ratio of the respective probability density functions, and compares it against a threshold which is designed such that a specified false alarm constraint is satisfied. 

In a first simple model for target detection it is assumed that the background clutter can be described by a statistical model in which the different range cells inside the sliding window contain statistically independent identically exponentially distributed (iid) random variables X i,..., X/v. The probability density function (pdf) of exponentially distributed clutter variables is fully described by the equation ... 

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. 

A theoretical framework based on the one-point, one-time joint probability density function (PDF) is developed. It is shown that all commonly employed models for turbulent reacting flows can be formulated in terms of the joint PDF of the chemical species and enthalpy. Models based on direct closures for the chemical source term as well as transported PDF methods, are covered in detail. An introduction to the theory of turbulence and turbulent scalar transport is provided for completeness. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

Dreeben, T. D. and S. B. Pope (1997a). Probability density function and Reynolds-stress modeling of near-wall turbulent flows. Physics of Fluids 9, 154—163. 

Lakatos, B. and E. Varga (1988). Probability density functions based micromixing model of chemical reactors. Computers in Chemical Engineering 12, 165-169. 

Wall, C., B. J. Boersma, and R Moin (2000). An evaluation of the assumed beta probability density function subgrid-scale model for large eddy simulation of nonpremixed, turbulent combustion with heat release. Physics of Fluids 12, 2522-2529. 

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