High-order particle method for solving incompressible Navier–Stokes equations within a mixed Lagrangian–Eulerian framework

Owing to the fact that the Poisson equation of pressure for incompressible fluid flow is purely elliptic, it is therefore computationally improper to compute pressure on irregularly distributed moving particles as addressed in the recent Moving Particle with embedded Pressure Mesh (MPPM) method. In the current work, a modified MPPM method known as the Mixed Lagrangian–Eulerian (MLE) method is proposed for solving the incompressible Navier–Stokes equations. In the current velocity–pressure formulation, the momentum and continuity equations are approximated on the moving particles (Lagrangian) and the uniform Cartesian grid points, respectively. Meanwhile, the total derivative of velocity terms appeared in the momentum equations are estimated by simply advecting the moving particles, thereby eliminating the convection stability problem and increasing the flow accuracy without introducing false diffusion error. In the conventional Moving Particle Semi-implicit (MPS) and MPPM methods, numerical accuracies of the Laplacian and gradient operators are strongly dependent on the regularity of the particle distribution. In some implicit schemes, the gradient and Laplacian terms are of second-order and first-order accuracy, respectively. In the current work, the second-order accuracies of these differential terms exhibited on moving particles are realized by interpolating the derivative values from the uniform Cartesian grids calculated by using the high-order Combined Compact Difference (CCD) scheme. From the numerical results of Laplacian term approximation by using various numerical schemes, it is shown that the new MLE scheme is at least second-order accurate. The proposed Mixed Lagrangian–Eulerian (MLE) method can be easily applied to simulate fluid flow problems ranging from low to high Reynolds number. It is found that the numerical results compare well with the benchmark solutions. Moreover, it is more accurate than the recently proposed MPPM method. •A mixed-Lagrangian–Eulerian (MLE) method is proposed.•The total derivative terms are solved in the Lagrangian sense.•The spatial derivative terms are solved in the Eulerian.•The continuity equation is solved on Cartesian mesh to retain the elliptic nature.•Data transfer between particles and grids are realized by interpolations.