An Eulerian-Lagrangian particle method for weakly compressible viscous flows using peridynamic differential operator

This study proposes a peridynamic differential operator (PDDO)-based Eulerian-Lagrangian hybrid particle method for weakly compressible viscous flows. The PDDO is utilized to transform the governing partial differential equations into their integral form; hence, the issue of local non-differentiability is eliminated. Both the Eulerian and Lagrangian formulations of particle methods for solving the governing integral equations are derived and then combined to propose the hybrid method. In the proposed method, the Eulerian and Lagrangian formulation is utilized to solve the governing equations in the inner computational domain and in areas with free surfaces, respectively. The results of these two domains are then merged to arrive at the solution. In such a way, the hybrid particle method effectively balances the computational demands and applicability. The numerical stability and interpolation consistency of the proposed method are shown by introducing the laminar viscosity model. A detailed numerical procedure is provided involving the boundary conditions and time-stepping strategy. The proposed method is validated by several benchmark problems. Furthermore, the method was applied to several benchmark problems including the hydrostatic test, Taylor-Green vortex, and numerical wave generation. The results of these numerical examples suggest that the proposed method is computationally less demanding while maintaining accuracy.