Conservation and accuracy studies of the LESCM for incompressible fluids

Conservation properties are very important for numerical methods. The presence of non-conservative solutions can result in fictitious sources or sinks that cause alterations in the flow balance. In this paper, the conservation properties and their influences on the accuracy of a newly proposed meshfree Lagrangian-Eulerian stabilized collocation method (LESCM) for the incompressible fluid flow are mathematically derived and numerically validated. The LESCM inspired by the material point method (MPM) is based on the hybrid Lagrangian-Eulerian description, in which the accuracy of the solution on Eulerian nodes and the mapping between Eulerian nodes and Lagrangian particles are highly associated with the conservation properties. On one hand, since the Navier-Stokes equations are solved without any truncation or simplification, the global conservation can be satisfied like most of the numerical methods during the solution on the Eulerian nodes. Moreover, this method can also meet the global conservation for the mass, linear momentum and angular momentum during the mapping process which are detailedly derived in the paper. After that, the energy conservation is evaluated by the corresponding equation. On the other hand, the local conservation properties of the discrete equations as well as the mass and momentums can be satisfied in the LESCM, while most of the numerical methods do not possess this property. Therefore, the conservation properties can be maintained during both the solving and the mapping processes which guarantee the high accuracy, optimal convergence and good stability of the LESCM method. Numerical examples of steady-state flow, shear-driven cavity, dam break and water sloshing problems with complex free surface and breaking waves, demonstrate the high accuracy and efficiency as well as conservation of the LESCM. It also performs very well for fluid flow problems with large Reynolds numbers. This method can be extensively applied to the engineering problems associated with the incompressible free surface flow.