The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi>δ</mml:mi></mml:math>-ALE-SPH model: An arbitrary Lagrangian-Eulerian framework for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"><mml:mi>δ</mml:mi></mml:math>-SPH model with particle shifting technique

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) rely use Riemann solvers inside spatial operators, proposed model expressed terms primitive density velocity) written using standard differential formulations schemes. Similarly to ALE-SPH models, velocity field modifying pure Lagrangian material point through δu→ given Particle Shifting Technique (PST). We show that above-mentioned are, however, unstable when they are integrated time. instability appears form large volume variations those fluid regions characterised high strain rates. Nonetheless, can be stabilised if appropriate diffusion included both mass. This latter scheme, hereinafter called δ-ALE-SPH validated against reference benchmark test-cases: viscous flow around inclined elliptical cylinder, lid-driven cavity dam-break impacting vertical wall.