The parabolic edge reconstruction method (PERM) for Lagrangian particle advection

We describe a Lagrangian particle advection scheme which is intended for use in hybrid finite-volume (FV) large-eddy simulation/filtered density function (LES/FDF) methods for low-Mach flows, but which may also be applicable to unsteady probability density function (PDF) methods, direct numerical simulation (DNS) or any other situation where tracking fluid particles is of concern. A key ingredient of the scheme is a subgrid reconstruction of the filtered velocity field with desirable divergence properties, which is necessary for accurate evolution of the particle number density. We develop reconstructions for 2D and 3D Cartesian staggered non-uniform grids. The reconstructed velocity field is continuous and piecewise parabolic in the velocity-component direction. In the direction normal to the velocity component the reconstruction is piecewise linear. The divergence of the reconstructed field is bilinear in 2D (trilinear in 3D) within a given cell and consistent with the discrete divergence given by the staggered-grid velocities. Though the reconstructed divergence field may be discontinuous from cell to cell, the norm of the differences between the vertex values of the reconstructed divergence for neighboring cells is minimized. As a consequence, the divergence is everywhere zero for the constant-density case. A two-stage Runge–Kutta scheme is employed for advancement of the particle positions. To assess the performance of the scheme we utilize a set of non-trivial velocity test functions which are designed to mimic realistic flow fields. We show that an advection scheme based on the new velocity reconstruction method is effective at maintaining an accurate particle number density in the particle-tracking limit.