A second-order in time and space particle-based method to solve flow problems on arbitrary meshes

Highlights: • A novel numerical method based on particles and finite volumes was developed. • High-order interpolation and projection operators were designed and tested on arbitrary meshes. • Quadratic space–time convergence was achieved on flow problems even using large time-steps. • A parallel implementation in the OpenFOAM library was performed and evaluated. • The methodology is shown to be more efficient than a fast Eulerian alternative. -- Abstract: This work presents a novel proposal of a second-order accurate (in time and space) particle-based method for solving transport equations including incompressible flows problems within a mixed Lagrangian–Eulerian formulation. This methodology consists of a symmetrical operator splitting, the use of high-order operators to transfer data between the particles and the background mesh, and an improved version of the eXplicit Integration Following the Streamlines (X–IVS) method. In the case of incompressible flows, a large time-step iterative solver is employed where the momentum equation is split to improve the numerical approximation of the convective term. New interpolation and projection operators are evaluated and quadratically accurate solutions of scalar transport tests are presented. Then, incompressible flow problems are solved where the rate of convergence of the method is assessed using both structured and unstructured background grids. The method is implemented in the open source platform OpenFOAM^®allowing employing arbitrary meshes and obtaining reliable computing time comparisons with standardized solvers. The results obtained reveal that the current method is able to obtain a lower level of error than a fast Eulerian alternative, without increasing the total computing time.