An Eulerian-Lagrangian incompressible SPH formulation (ELI-SPH) connected with a sharp interface

An Eulerian-Lagrangian incompressible SPH (ELI-SPH) formulation is proposed that improves accuracy over a fully Lagrangian formulation for many problems. This develops the original formulation of Lind and Stansby (2016) by providing a sharp interface rather than a transition zone. This is generally convenient and avoids any need for Arbitrary Lagrangian Eulerian (ALE) particle mass correction. It also enables a simple, accurate solid boundary condition in a Lagrangian formulation by having the interface close to the solid boundary with an Eulerian fluid domain typically three particles thick with mirror particles. Particle regularisation is necessary in a Lagrangian domain and we apply a general form based on Fick's shifting which is modified at the interface by ignoring Eulerian particles and using mirror particles to give zero concentration gradient and hence zero shifting across the interface, avoiding spurious migration. Continuity is enforced at the interface as part of the combined Eulerian-Lagrangian domain. The formulation is validated against the analytical solution for Taylor-Green vortices, vortex spin down in a box, and propagating waves. The use of mixed kernel order in the Eulerian domain is also demonstrated.