An analysis of the convergence of stochastic Lagrangian/Eulerian spray simulations

Abstract

This work derives how the convergence of stochastic Lagrangian/Eulerian simulations depends on the number of computational parcels, particularly for the case of spray modeling. A new, simple, formula is derived that can be used for managing the numerical error in two or three dimensional computational studies. For example, keeping the number of parcels per cell constant as the mesh is refined yields an order one-half convergence rate in transient spray simulations. First order convergence would require a doubling of the number of parcels per cell when the cell size is halved. Second order convergence would require increasing the number of parcels per cell by a factor of eight. The results show that controlling statistical error requires dramatically larger numbers of parcels than have typically been used, which explains why convergence has been so elusive.

Large-eddy spray simulation under direct-injection spark-ignition engine-like conditions with an integrated atomization/breakup model

“…Some studies have argued that the number of injected parcels should increase as the mesh refines in order to achieve convergence. 68,69 However, the number of injected parcels in this work is comparable to the one reported in Senecal et al 69 for the same level of mesh refinement. As will be discussed later, the number of injected parcels is sufficient that running multiple realizations has little impact on the global spray characteristics such as liquid and vapor penetrations.…”

Comprehensive Analysis of the Impact of a Novel Droplet Volume Fraction-Based Drag Reduction Correlation in a Numerical Study on Water Sprays with Different Levels of Density

“…In 2006, Schmidt stated that if the number of computational particles does not increase while the cell size is reduced, the mean expected error increases after having reached an optimum [27]. As shown by Schmidt et al [28], the most challenging part in the Lagrangian-Eulerian computation is the 'liquid to gas coupling', or in other words, the accumulation of the mass, momentum, energy and species contributions of the dispersed particles on the source terms in the gas phase equations. As a consequence of the limited number of tracked particles, there is a statistical uncertainty in the source terms.…”

Effect of the scale resolution on the two phase coupling characteristics of high speed evaporating sprays using LES / Eulerian-Lagrangian methodologies

“…For example in the recent work of Schmidt and Bedford, 2018 a framework for Eulerian-Lagrangian simulations has been suggested that controls the number of parcels used in order to succeed convergence.…”