An LES-PBE-PDF approach for modeling particle formation in turbulent reacting flows

Abstract

Many chemical and environmental processes involve the formation of a polydispersed particulate phase in a turbulent carrier flow. Frequently, the immersed particles are characterized by an intrinsic property such as the particle size and the distribution of this property across a sample population is taken as an indicator for the quality of the particulate product or its environmental impact.In the present article, we propose a comprehensive model and an efficient numerical solution scheme for predicting the evolution of the property distribution associated with a polydispersed particulate phase forming in a turbulent reacting flow. Here, the particulate phase is described in terms of the particle number density whose evolution in both physical and particle property space is governed by the population balance equation (PBE). Based on the concept of large eddy simulation (LES), we augment the existing LES-transported PDF approach for fluid phase scalars by the particle number density and obtain a modelled evolution equation for the filtered probability density function (PDF) associated with the instantaneous fluid composition and particle property distribution. This LES-PBE-PDF approach allows us to predict the LES-filtered fluid composition and particle property distribution at each spatial location and point in time without any restriction on the chemical or particle formation kinetics. In view of a numerical solution, we apply the method of Eulerian stochastic fields, invoking an explicit adaptive grid technique in order to discretize the stochastic field equation for the number density in particle property space. In this way, sharp moving features of the particle property distribution can be accurately resolved at a significantly reduced computational cost.As a test case, we consider the condensation of an aerosol in a developed turbulent mixing layer. Our investigation not only demonstrates the predictive capabilities of the LES-PBE-PDF model, but also indicates the computational efficiency of the numerical solution scheme.

Effect of soot model, moment method, and chemical kinetics on soot formation in a model aircraft combustor

“…Improvements to the statistical approaches that can ensure numerical convergence of the moments approach will be useful for ensuring reliability of the predictions. In this sense, other solution techniques [28,29] show promise and should be further investigated. From a physics standpoint, the performance of such models in combustors where nucleation/condensation dominate soot mass would be interesting, and the hypothesis is that differences due to chemical mechanisms would be larger but differences due to statistical models smaller.…”

“…Similarly, the description of the soot population can be achieved through multiple techniques [2,12,[27][28][29] . In general, the soot population is described in terms of a number density function (NDF), which is a spatially and temporally varying function of certain internal coordinates that describe particle characteristics.…”

An LES-PBE-PDF approach for predicting the soot particle size distribution in turbulent flames

“…Formally, the fractional time stepping constitutes a first order approximation in time [58]; its main benefit is that the numerical solution schemes can be tailored to the physical processes which they target. The stochastic field equation associated with the particle number density, moreover, is discretized in particle size space using the explicit adaptive grid method recently developed by Sewerin and Rigopoulos [68] in combination with a high resolution finite volume method [36], also see Sewerin and Rigopoulos [69]. Here, the grid design is controlled by three parameters, the total number of nodes, the minimum node density in the nucleation interval and the maximum grid stretching, which we set to 30, 4 nodes/4.75 nm and 2, respectively.…”

“…Here, the grid design is controlled by three parameters, the total number of nodes, the minimum node density in the nucleation interval and the maximum grid stretching, which we set to 30, 4 nodes/4.75 nm and 2, respectively. Similar to the application of a fractional steps scheme to the stochastic equations governing the gas phase scalars, the stochastic field equation for the (transformed) number density is decomposed into a convection/diffusion, molecular mixing and PBE fractional step [68,69].…”

“…From a physical perspective, the MMC-LES approach is based on shifting micromixing locality from physical space to the space spanned by the reactive scalars and discrete number densities. Sewerin and Rigopoulos [69], on the other hand, revisited the PBE-PDF rationale and formulated a joint scalar-number density pdf transport equation that is independent of a specific particle size discretization. An important consequence is that particle size persists as an independent coordinate in a statistically equivalent stochastic particle or field reformulation and that these stochastic systems may hence be discretized with respect to particle size using both fixed and adaptive grid schemes.…”