Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

Accurate, non-oscillatory, remeshing schemes for particle methods

“…However the particle method uses significantly less particles than in [12] and the time step is larger. Our number of particles is comparable to that in [25] but our time step is significantly larger. The higher resolution case is shown to illustrate the resolution necessary to prevent the lack of connectivity of the surface.…”

“…The computations are done in a unit cubic box.The sphere is initially centered at the point of coordinates (0.5, 0.5, 0.5) and its radius is 0.15. In Figure 7 we compare the particle method using the corrected Λ 4 scheme and a second order time splitting, with several similar recent experiments in the literature where lagrangian markers are used to complement grid-based methods: the particle level set method [12] and the volume of fluid method [25]. To compare with [12,25] we have first considered the following velocity field      u(x, y, z, t) = 2f (t) sin 2 (πx) sin(2πy) sin(2πz), v(x, y, z, t) = −f (t) sin(2πx) sin 2 (πy) sin(2πz), w(x, y, z, t) = −f (t) sin(2πx) sin(2πy) sin 2 (πz) (43) with f (t) = cos(πt/T ), T = 3.…”

“…The time-step value is not mentioned in the reference, but the CFL number, related to the finite-difference grid spacing is presumably of the order of 1. The top right picture shows the results of [25]. In this reference a Volume of Fluid method is used on a 64 3 grid, and complemented with nine markers per cell, and the CFL number was 0.1.…”

“…The CFL number resulting from (13) [12] with N = 100 grid points complemented with 4 particles per grid-size in each direction around the interface. Top-right : [25], N = 64 and 9 particles per grid-cell, CF L = 0.1. Bottom : present 4th order method with N = 100 and CF L = 8 (left picture), N = 160 and CF L = 12 (right picture).…”

“…In [12] subgrid particles are introduced near the interface to improve the accuracy of finite-difference level set methods. [25] describes a Volume-of-Fluid method for advection problems designed in the same spirit for general transport equations. These works take advantage of the exact Lagrangian solution to advection equation in order to limit the numerical diffusion of grid-based method and enforce mass conservation.…”