An Analysis of the Convergence of Stochastic Lagrangian/Eulerian Spray Simulations

TL;DR

This paper derives a formula linking the convergence rate of stochastic spray simulations to the number of parcels, highlighting the need for many parcels to achieve accurate results and explaining previous convergence challenges.

Contribution

It introduces a simple formula for managing numerical error in stochastic spray simulations and clarifies how parcel count affects convergence rates.

Findings

Order one-half convergence with constant parcels per cell as mesh refines

Doubling parcels per cell achieves first-order convergence

Increasing parcels by a factor of eight achieves second-order convergence

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.