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Solution of the Fokker-Planck Equation Using the Extended Quadrature Method of Moments
Abstract
One of the important attractions of employing the transported probability density function (PDF) methods for turbulent reacting flows is the fact that the chemical source terms are treated exactly. The composition probability density function (PDF) evolves with convective transport in physical space due to the mean velocity (macromixing), the turbulent diffusivity which transports the composition PDF in physical space (mesomixing), the transport in composition space due to molecular mixing (micromixing) and chemical reactions. The key model term of our interest in this study in the transport PDF equation is the molecular mixing term, which describes how molecular diffusion affects both the shape of the PDF and the rate of scalar variance decay. Molecular mixing is expressed using the Fokker-Planck model (Fox, 2003, 1994), which is an extension of the interaction-by-exchange-with-the-mean (IEM) model. The IEM model, widely used in chemical-reaction engineering and computational combustion due to its simple form, assumes the linear relaxation of the scalar towards its mean, while the Fokker-Plank model considers, in addition, the effect of the differential diffusion process to mimic the mixing. Differential diffusion occurs when the molecular diffusivities of the scalar fields are not the same. An extended quadrature-based moment method (EQMOM) (Yuan et al., 2012; Chalons C. Fox R.-O. Massot M., 2010) is used to close the transport equation for the composition PDF. The β kernel density function is used for EQMOM since the scalar composition is represented by a scalar bounded between -1 and 1. The PDF predicted by the EQMOM model in the same conditions studied in the direct numerical simulation of Eswaran and Pope (1988) are reported and compared to the DNS predictions.
References:
Chalons, C., Fox, R. O., Massot, M., 2010. A multi-Gaussian quadrature method of moments for gas-particle flows in a LES framework. Studying Turbulence Using Numerical Simulation Databases, Center for Turbulence Research, Summer Program 2010, Stanford University 347–358.
Eswaran, V., Pope, S.B., 1988. Direct numerical simulations of the turbulent mixing of a passive scalar. Physics of Fluids 31, 506–520.
Fox, R.O., 2003. Computational Models for Turbulent Reacting Flows. Cambridge University Press.
Fox, R.O., 1994. Improved Fokker–Planck model for the joint scalar, scalar gradient PDF. Physics of Fluids 6, 334–348.
Yuan, C., Laurent, F., Fox, R.O., 2012. An extended quadrature method of moments for population balance equations. Journal of Aerosol Science 51, 1–23.
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An extended quadrature method of moments (EQMOM) with log-normal kernel density functions is developed in this work, and applied to the solution of a population balance equation (PBE) for aggregation and breakup, coalescence, and condensation problems. The cases with one and two kernel density functions are studied analytically, and the existence of an analytical solution is shown. A numerical procedure based on the work of Yuan et al. (2012) is adopted to address cases with a larger number of kernel density functions. Results for the reconstructed number density function (NDF), the time evolution of the zero-order moment and of the mean particle size are compared with those obtained from the rigorous solution of the PBE reported by Vanni (2000) for the cases of aggregation and breakup. A problem concerning coalescence and one regarding condensation, both with analytical solution, are also examined. The results obtained with the proposed approach are compared to those provided by EQMOM with gamma kernel densities. Satisfactory results were obtained for the reconstructed distribution. Excellent agreement was observed between the rigorous solution and the approximated one for the time evolution of the total number density and the mean particle size.
The radiative transfer equation describes the propagation of radiation through a material medium. While it provides a highly accurate description of the radiation field, the large phase space on which the equation is defined makes it numerically challenging. As a consequence, significant effort has gone into the development of accurate approximation methods. Recently, an extended quadrature method of moments (EQMOM) has been developed to solve univariate population balance equations, which also have a large phase space and thus face similar computational challenges. The distinct advantage of the EQMOM approach over other moment methods is that it generates moment equations that are consistent with a positive phase space density and has a moment inversion algorithm that is fast and efficient. The goal of the current paper is to present the EQMOM method in the context of radiation transport, to discuss advantages and disadvantages, and to demonstrate its performance on a set of standard one-dimensional benchmark problems that encompass optically thin, thick, and transition regimes. Special attention is given in the implementation to the issue of realizability—that is, consistency with a positive phase space density. Numerical results in one dimension are promising and lay the foundation for extending the same framework to multiple dimensions.
Population balance equations (PBE) for a number density function (NDF) arise in many applications of aerosol technology. Thus, there has been considerable interest in the development of numerical methods to find solutions to PBE, especially in the context of spatially inhomogeneous systems where moment realizability becomes a significant issue. Quadrature-based moment methods (QBMM) are an important class of methods for which the accuracy of the solution can be improved in a controlled manner by increasing the number of quadrature nodes. However, when a large number of nodes is required to achieve the desired accuracy, the moment-inversion problem can become ill-conditioned. Moreover, oftentimes pointwise values of the NDF are required, but are unavailable with existing QBMM. In this work, a new generation of QBMM is introduced that provides an explicit form for the NDF. This extended quadrature method of moments (EQMOM) approximates the NDF by a sum of non-negative weight functions, which allows unclosed source terms to be computed with great accuracy by increasing the number of quadrature nodes independent of the number of transported moments. Here, we use EQMOM to solve a spatially homogeneous PBE with aggregation, breakage, condensation, and evaporation terms, and compare the results with analytical solutions whenever possible. However, by employing realizable finite-volume methods, the extension of EQMOM to spatially inhomogeneous systems is straightforward.
The evolution of scalar fields, of different initial integral length scales, in statistically stationary, homogeneous, isotropic turbulence is studied. The initial scalar fields conform, approximately, to 'double-delta function' probability density functions (pdf's). The initial scalar-to-velocity integral length-scale ratio is found to influence the rate of the subsequent evolution of the scalar fields, in accord with experimental observations of Warhaft and Lumley (1978). On the other hand, the pdf of the scalar is found to evolve in a similar fashion for all the scalar fields studied; and, as expected, it tends to a Gaussian. The pdf of the logarithm of the scalar-dissipation rate reaches an approximately Gaussian self-similar state. The scalar-dissipation spectrum function also becomes self-similar. The evolution of the conditional scalar-dissipation rate is also studied. The consequences of these results for closure models for the scalar pdf equation are discussed.
Direct Numerical Simulations (DNS) are performed of a homogeneous turbulent flow under the influence of a chemical reaction of the type A + B→ Products. The generated results are statistically analyzed to describe the process of mixing in an analogous plug flow reactor configuration. The results of the statistical analysis indicate that the Probability Density Functions (PDF's)of a conserved Shvab-Zeldovich variable, characterizing the mixing process, can be well approximated by a Beta distribution. This provides a justification for implementing this density for predicting the limiting bounds of the unmixedness in homogeneous reacting turbulence.
The method of moments (MOM) may be used to determine the evolution of the lower-order moments of an unknown aerosol distribution. Previous applications of the method have been limited by the requirement that the equations governing the evolution of the lower-order moments be in closed form. Here a new approach, the quadrature method of moments (QMOM), is described. The dynamical equations for moment evolution are replaced by a quadrature-based approximate set that satisfies closure under a much broader range of conditions without requiring that the size distribution or growth law maintain any special mathematical form. The conventional MOM is recovered as a special case of the QMOM under those conditions, e.g., free-molecular growth, for which conventional closure is satisfied. The QMOM is illustrated for the growth of sulfuric acid-water aerosols and simulations of diffusion-controlled cloud droplet growth are presented. (C) 1997 American Association for Aerosol Research.
Results are presented of direct numerical simulations (DNS) of spatially developing shear flows under the influence of infinitely fast chemical reactions of the type A + B yields Products. The simulation results are used to construct the compositional structure of the scalar field in a statistical manner. The results of this statistical analysis indicate that the use of a Beta density for the probability density function (PDF) of an appropriate Shvab-Zeldovich mixture fraction provides a very good estimate of the limiting bounds of the reactant conversion rate within the shear layer. This provides a strong justification for the implementation of this density in practical modeling of non-homogeneous turbulent reacting flows. However, the validity of the model cannot be generalized for predictions of higher order statistical quantities. A closed form analytical expression is presented for predicting the maximum rate of reactant conversion in non-homogeneous reacting turbulence.
Investigation of particulate systems often requires the solution of a population balance, which is a continuity statement written in terms of the number density function. In turn, the number density function is defined in terms of an internal coordinate (e.g., particle length, particle volume) and it generates integral and derivative terms. Different methods exist for numerically solving the population balance equation. For many processes of industrial significance, due to the strong coupling between particle interactions and fluid dynamics, the population balance must be solved as part of a computational fluid dynamics (CFD) simulation. Such an approach requires the addition of a large number of scalars and the associated transport equations. This increases the CPU time required for the simulation, and thus it is clear that it is very important to use as few scalars as possible. In this work the quadrature method of moments (QMOM) is used. The QMOM has already been validated for crystal growth and aggregation; here the method is extended to include breakage. QMOM performance is tested for 10 different cases in which the competition between aggregation and breakage leads to asymptotic solutions.
ISU) EQMOM for Fokker-Planck Equation
- E Madadi
- A Passalacqua
E. Madadi and A. Passalacqua (ISU)
EQMOM for Fokker-Planck Equation
November 10 th, 2015
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