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The evolution of polydisperse systems is governed by population balance equations. A group of efficient solution approaches are the moment methods, which do not solve for the number density function (NDF) directly but rather for a set of its moments. While this is computationally efficient, a specific challenge arises when describing the fluxes across a boundary in phase space for the disappearance of elements, the so-called zero-flux. The main difficulty is the missing NDF-information at the boundary, which most moment methods cannot provide. Relevant physical examples are evaporating droplets, soot oxidation or particle dissolution.
In general, this issue can be solved by reconstructing the NDF close to the boundary. However, this was previously only achieved with univariate approaches, i.e. considering only a single internal variable. Many physical problems are insufficiently described by univariate population balance equations, e.g. droplets in sprays often require the temperature or the velocity to be internal coordinates in addition to the size.
In this paper, we propose an algorithm, which provides an efficient multivariate approach to calculate the zero-fluxes. The algorithm employs the Extended Quadrature Method of Moments (EQMOM) with Beta and Gamma kernel density functions for the marginal NDF reconstruction and a polynomial or spline for the other conditional dimensions. This combination allows to reconstruct the entire multivariate NDF and based on this, expressions for the disappearance flux are derived. An algorithm is proposed for the whole moment inversion and reconstruction process. It is validated against a suite of test cases with increasing complexity. The influence of the number of kernel density functions and the configuration of the polynomials and splines on the accuracy is discussed. Finally, the associated computational costs are evaluated.
Numerical simulation of a gasoline spray using one-dimensional turbulence for primary atomization
(2018)
Predictive and reliable simulations have the potential to constitute a valuable tool for the optimization of spray systems if accurate submodels are developed for the entire range of the governing processes. The primary breakup of the turbulent liquid jet is one the most important mechanisms in sprays, yet the least developed in terms of numerical modeling. The most accurate method to simulate primary breakup is the proper resolution of liquid-gas interfaces and turbulent flow structures. However, a wide range of relevant length and time scales implicate grid requirements that are often prohibitive for real engineering applications. The most widely used method in practice is still the representation of both the continuous liquid core and the dispersed phase by means of discrete Lagrangian particles evolving in and interacting with the Eulerian gas phase. The available models for primary breakup are mainly phenomenological and involve a number of empirical constants. The one-dimensional turbulence (ODT) model is an alternative stochastic approach to model turbulence in flows with a dominant direction of property gradients. The stochastic representation of turbulent eddies on a one-dimensional domain enables high resolution at moderate computational costs. Applications of ODT to atomization revealed a great potential in recent studies. The objective of the present study is to combine ODT as a primary breakup model with a conventional Eulerian-Lagrangian method for the further spray evolution in order to asses ODT as a submodel in full spray models. Our numerical investigations were conducted on the ECN spray G, a gasoline-like, evaporating spray. The results in terms of spray penetration are encouraging, though the applicability of ODT to the transient injection phase and effects on additional spray characteristics require further investigation.