Probability density function method

Anand, M. S., A. T. Hsu, and S. B. Pope (1997). Calculations of swirl combustors using joint velocity-scalar probability density function methods.. 47.4.4 Journal 35, 1143-1150. 

Calculations of subsonic and supersonic turbulent reacting mixing layers using probability density function methods. Physics of Fluids 10, 497 498. 

Wall-boundary conditions in probability density function methods and application to a turbulent channel flow. Physics of Fluids 11, 2632-2644. 

Raju, M. S. (1996). Application of scalar Monte Carlo probability density function method for turbulent spray flames. Numerical Heat Transfer, Part A 30, 753-777. 

The Presumed Probability Density Function method is developed and implemented to study turbulent flame stabilization and combustion control in subsonic combustors with flame holders. The method considers turbulence-chemistry interaction, multiple thermo-chemical variables, variable pressure, near-wall effects, and provides the efficient research tool for studying flame stabilization and blow-off in practical ramjet burners. Nonreflecting multidimensional boundary conditions at open boundaries are derived, and implemented into the current research. The boundary conditions provide transparency to acoustic waves generated in bluff-body stabilized combustion zones, thus avoiding numerically induced oscillations and instabilities. It is shown that predicted flow patterns in a combustor are essentially affected by the boundary conditions. The derived nonreflecting boundary conditions provide the solutions corresponding to experimental findings. 

Macro- and Micro-Mixing in Reactors Models Explicitly Accounting for Mixing Micro-Probability Density Function Methods... 

State vector, specification of, 493 Stationarity property of probability density functions, 136 Stationary methods, 60 Statistical independence, 148 Statistical matrix, 419 including description of "mixtures, 423... 

Table 2.3 is used to classify the differing systems of equations, encountered in chemical reactor applications and the normal method of parameter identification. As shown, the optimal values of the system parameters can be estimated using a suitable error criterion, such as the methods of least squares, maximum likelihood or probability density function. 

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-... 

These differential equations depend on the entire probability density function / (x, t) for x(t). The evolution with time of the probability density function can, in principle, be solved with Kolmogorov s forward equation (Jazwinski, 1970), although this equation has been solved only in a few simple cases (Bancha-Reid, 1960). The implementation of practical algorithms for the computation of the estimate and its error covariance requires methods that do not depend on knowing p(x, t). 

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. 

---