Notes on the SPH model within the Arbitrary-Lagragian-Eulerian framework

In the present work we study the behaviour of a weakly-compressible SPH scheme obtained by rewriting the Navier-Stokes equations in an Arbitrary-Lagragian-Eulerian (ALE) format. The theoretical framework is that described in Vila (1999) [1], even though the proposed SPH model is expressed in terms of primitive variables (i.e. density and velocity) instead of conservative ones. Differently from the classical approach to ALE, which is based on the use of Riemann solvers inside the spatial operators (see for details [1]), the present model is written by using the standard differential formulations of the weakly-compressible SPH schemes. Similarly to Oger et al. (2016) [2], the arbitrary velocity field is obtained by modifying the pure Lagrangian velocity of the material point through a velocity ?u given by a Particle Shifting Technique (PST). We show that the above-mentioned ALE-SPH equations are, however, unstable when they are integrated in time. The instability appears in the form of large volume variations in those fluid regions characterised by high velocity strain rates. Nonetheless, the scheme can be stabilised if appropriate diffusion terms are included in both the equations of density and mass. This latter scheme, hereinafter called ?-ALE-SPH scheme, is validated against a reference benchmark test-case: the viscous flow around an inclined elliptical cylinder.