Relaxation Model for Pressureless Air-Mixed Droplet Flow in Atmospheric Icing

This study addresses numerical challenges in simulating pressureless air-mixed droplet flows under atmospheric icing conditions. The governing equations, commonly referred to as the Eulerian droplet equations, comprise a combination of the pressureless gas dynamic (PGD) equations and source terms that account for drag and gravitational effects. Within this framework, three fundamental challenges are identified: 1) the need for a relaxation model to handle the advection terms, 2) numerical instabilities induced by stiff source terms, and 3) the formulation of suitable boundary conditions on solid surfaces. For these challenges, we propose novel numerical strategies. First, we introduce a relaxation model that retains mathematical equivalence with the PGD equations, facilitating the implementation of a Godunov-type upwind scheme for spatial flux computation. Second, we employ a time integration scheme for the source terms based on ordinary differential equation solvers, enhancing numerical stability through the introduction of a Courant–Friedrichs–Lewy (CFL)-like parameter. Third, we present characteristic equation-based boundary conditions to accurately capture droplet impingement behavior and reduce spurious peaks in collection efficiency near stagnation points. In addition, a positivity-preserving intercell flux formulation is implemented to ensure non-negative liquid water content in shadow regions. Validation results demonstrate that the proposed approaches improve both the accuracy and stability of numerical simulations of pressureless air-mixed droplet flows in icing environments.