Probability density function PDF method

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-... 

The molecular dynamics, Monte Carlo methods, the Lagrangian probability density function (PDF) methods, and the Lattice Boltzmann (LBM) method are among these methods. Methods such as the smoothed particle hydrodynamics (SPH) and the vortex method initially developed as probabiUstic methods, but nowadays they are most frequently used as deterministic. 

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. 

The present study is to elaborate on the computational approaches to explore flame stabilization techniques in subsonic ramjets, and to control combustion both passively and actively. The primary focus is on statistical models of turbulent combustion, in particular, the Presumed Probability Density Function (PPDF) method and the Pressure-Coupled Joint Velocity-Scalar Probability Density Function (PC JVS PDF) method [23, 24]. 

The deterministic approach of direct numerical simulation (DNS) and the probabilistic approach of probability density function (PDF) modeling are implemented for prediction of droplet dispersion and polydis-persity in liquid-fuel combustors. For DNS, a multidomain spectral element method was used for the carrier phase while tracking the droplets individually in a Lagrangian frame. The geometry considered here is a backward-facing step flow with and without a countercurrent shear. In PDF modeling, the extension of previous work to the case of evaporating droplets is discussed. 

The bisection method (Raiffa 1968) is another direct technique for attempting to estimate a subjective probability density function (pdf). This technique is somewhat more general than fitting the subject s belief with a functional form, such as the beta-1, since it makes no parametric assumptions. The bisection method involves two steps which arc repeated imtil the subject s belief is adequately described. Following this approach, the first step is to determine the median (Pos) of the subjective pdf. This question is posed to the decision maker in a form such as For what value of p do you feel it is equally likely the true value pt is greater than or less than p This step is then repeated for subintervals to obtain the desired level of detail. 

The Bayesian time-domain approach presented in this chapter addresses this problem of parametric identification of linear dynamical models using a measured nonstationary response time history. This method has an explicit treatment on the nonstationarity of the response measurements and is based on an approximated probability density function (PDF) expansion of the response measurements. It allows for the direct calculation of the updated PDF of the model parameters. Therefore, the method provides not only the most probable values of the model parameters but also their associated uncertainty using one set of response data only. It is found that the updated PDF can be well approximated by an appropriately selected multi-variate Gaussian distribution centered at the most probable values of the parameters if the problem is... 

The remainder of the chapter focuses on the actual spray modeling. The exposition is primarily done for the RANS method, but with the indicated modifications, the methodology also applies to LES. The liquid phase is described by means of a probability density function (PDF). The various submodels needed to determine this PDF are derived from drop-drop and drop-gas interactions. These submodels include drop collisions, drop deformation, and drop breakup, as well as drop drag, drop evaporation, and chemical reactions. Also, the interaction between gas phase, liquid phase, turbulence, and chemistry is examined in some detail. Further, a discussion of the boundary conditions is given, in particular, a description of the wall functions used for the simulations of the boundary layers and the heat transfer between the gas and its confining walls. 

Whereas standard linearized localization methods give a single point solution and uncertainty estimates (Schechinger and Vogel 2005), die result of the nonlinear method is a probability density function (PDF) over the unknown source coordinates. The optimal location is taken as the maximum likelihood point of the PDF. The PDF explicitly accounts for a priori known data errors, which are assumed to be Gaussian. 

---