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 δ 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 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.

中文翻译：

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δ-ALE-SPH 模型：具有粒子移动技术的 δ-SPH 模型的任意拉格朗日-欧拉框架

摘要 研究了通过以任意拉格朗日-欧拉 (ALE) 格式重写 Navier-Stokes 方程而获得的弱可压缩 SPH 方案的行为。不同于以前的 ALE 工作，它通常采用保守变量（即质量和动量）并依赖于在空间算子内部使用黎曼求解器，所提出的模型以原始变量（即密度和速度）表示并写成通过使用弱可压缩 SPH 方案的标准微分公式。与 ALE-SPH 模型类似，任意速度场是通过通过粒子移动技术 (PST) 给出的速度 δ u → 修改材料点的纯拉格朗日速度来获得的。我们表明，上述 ALE-SPH 方程在时间积分时是不稳定的。在以高速应变率为特征的那些流体区域中，不稳定性以大体积变化的形式出现。尽管如此，如果在密度和质量方程中都包含适当的扩散项，则该方案可以稳定。后一种方案，以下称为 δ -ALE-SPH 方案，针对参考基准测试案例进行了验证：倾斜椭圆圆柱周围的粘性流、盖子驱动的腔体和冲击垂直壁的堤坝溃坝流。