A pressure-projection formulation in a least-squares meshfree method for the incompressible Navier-Stokes equations using a staggered-variable arrangement

Abstract

Incompressible flow solvers based on strong-form meshfree methods represent arbitrary geometries without the need for a global mesh system. However, their local evaluations make it difficult to satisfy incompressibility at the discrete level. Moreover, the collocated arrangement of velocity and pressure variables tends to induce a zero-energy mode, leading to decoupling between the two variables. In projection-based approaches, a spatial discretization scheme based on a conventional node-based moving least-squares method for the pressure causes inconsistency between the discrete operators on both sides of the Poisson equation. Thus, a solenoidal velocity field cannot be ensured numerically. In this study, a numerical method for the incompressible Navier-Stokes equations is developed by introducing a local primal-dual grid into the mesh-constrained discrete point method, enabling consistent discrete operators. The constructed virtual dual cell is defined solely from the local connectivity among nodes, and thus the method retains its meshfree nature. To achieve a consistent coupling between velocity and pressure variables under the primal-dual arrangement, time evolution converting is applied to evolve the velocity on cell interfaces. For numerical validation, a linear acoustic equation is solved to confirm the effectiveness of the staggered-variable arrangement based on the local primal-dual grid. Then, incompressible Navier-Stokes equations are solved, and the proposed method is demonstrated to ensure a local divergence-free velocity field up to an arbitrarily small discrete error, achieve the expected spatial convergence order, and accurately reproduce flow features over a wide range of Reynolds numbers.