A point-mass particle method for the simulation of immiscible multiphase flows on an Eulerian grid

We present an Eulerian-Lagrangian approach for the modeling and simulation of immiscible multiphase flow systems. The Naiver-Stokes equations are solved on a traditional Eulerian grid while the fluid mass and phase information is discretized by Lagrangian particles. The method is novel because the particles move with a velocity that enforces consistency between the particle field density and the fluid density. The approach is advantageous in that (i) an arbitrary number of phases are easily represented, (ii) the particles remain well-distributed in space, even near merging and diverging characteristics, (iii) mass conservation is easily controlled, and (iv) the methodology is applicable to a wide range of Courant numbers. The governing equations are derived and a numerical method is presented that is applicable to incompressible flows. Performance is assessed via standard two-dimensional and three-dimensional phase transport tests as a function of both Eulerian grid resolution and Lagrangian particle resolution. Results show that the shape error converges with first-order with respect to increasing either Eulerian grid resolution or particle resolution, while mass conservation errors converge with the square root. The method is shown to successfully simulate expanding elliptical regions, stationary and oscillating droplets, a droplet in shear flow, a Rayleigh-Taylor instability, and the air blast atomization of a droplet.