The [formula omitted]-ALE-SPH model: An arbitrary Lagrangian-Eulerian framework for the [formula omitted]-SPH model with particle shifting technique

Abstract

The behaviour of a weakly-compressible SPH scheme obtained by rewriting the Navier-Stokes equations in an arbitrary Lagrangian-Eulerian (ALE) format is studied. Differently from previous works on ALE, which generally adopt conservative variables (i.e. mass and momentum) and rely on the use of Riemann solvers inside the spatial operators, the proposed model is expressed in terms of primitive variables (i.e. density and velocity) and is written by using the standard differential formulations of the weakly-compressible SPH schemes. Similarly to ALE-SPH models, the arbitrary velocity field is obtained by modifying the pure Lagrangian velocity of the material point through a velocity δu→ 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 δ-ALE-SPH scheme, is validated against reference benchmark test-cases: the viscous flow around an inclined elliptical cylinder, the lid-driven cavity and a dam-break flow impacting a vertical wall.