Numerical Stability and Accuracy Considerations for the Solution of Lagrangian Stochastic Particle Dispersion Model Equations

Based on the classical Langevin equation for molecular diffusion, Lagrangian stochastic dispersion models are a popular method for describing the turbulent diffusion of particles in the atmospheric boundary layer. Pioneering work in the 1980's led to the formulation of a consistent framework for the formulation of Langevin-based Lagrangian stochastic dispersion models. Although such models are theoretically consistent in that they agree with their corresponding macroscopic Eulerian equation, they have been commonly observed to break down when applied to flows that contain substantial inhomogeneity or anisotropy. Modeled particle plumes have been reported to violate the second law of thermodynamics (i.e., entropy increases), and contain “rogue” particles whose velocities approach infinity. As a result, some type of ad hoc intervention is frequently needed to eliminate unphysical behavior, which can also lead to questionable results. The present work identifies the source of such unphysical model behavior, and presents a straightforward remedy that is consistent with theory. It was found that “rogue” trajectories are associated with numerical stability, while numerical accuracy dictates the degree to which the second law of thermodynamics or “well-mixed condition” is satisfied. Simple verification tests are suggested that future workers can implement to ensure computed particle plumes not only satisfy the second law of thermodynamics, but more fundamentally that they agree with Eulerian velocity statistics specified as inputs. Several test cases are presented that demonstrate model use in both Reynolds-averaged and large-eddy simulation applications.