A highly accurate SPH method with a new stabilization paradigm has been introduced by the authors in a recent paper aimed to solve Euler equations for ideal gases. We present here the extension of the method to viscous incompressible flow. Incompressibility is tackled assuming a weakly compressible approach. The method adopts the SPH-ALE framework and improves accuracy by taking high-order variable reconstruction of the Riemann states at the midpoints between interacting particles. The moving least squares technique is used to estimate the derivatives required for the Taylor approximations for convective fluxes, and also provides the derivatives needed to discretize the viscous flux terms. Stability is preserved by implementing the a posteriori Multi-dimensional Optimal Order Detection (MOOD) method procedure thus avoiding the utilization of any slope/flux limiter or artificial viscosity. The capabilities of the method are illustrated by solving one- and two-dimensional Riemann problems and benchmark cases. The proposed methodology shows improvements in accuracy in the Riemann problems and does not require any parameter calibration. In addition, the method is extended to the solution of viscous flow and results are validated with the analytical Taylor–Green, Couette and Poiseuille flows, and lid-driven cavity test cases.

中文翻译：

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具有后验稳定的弱可压缩粘性流的SPH-ALE方案

作者在最近的一篇论文中介绍了一种具有新稳定范式的高精度SPH方法，该方法旨在求解理想气体的欧拉方程。我们在这里介绍了该方法对粘性不可压缩流的扩展。假设采用弱压缩方法，将解决不可压缩性。该方法采用SPH-ALE框架，并通过在相互作用粒子之间的中点对Riemann态进行高阶变量重构来提高精度。移动最小二乘技术用于估计对流通量的泰勒近似所需的导数，并且还提供离散化粘性通量项所需的导数。通过实施后验多维最优顺序检测（MOOD）方法程序可保持稳定性，从而避免使用任何斜率/通量限制剂或人工粘度。通过解决一维和二维黎曼问题以及基准案例，说明了该方法的功能。所提出的方法显示了在黎曼问题中准确性的提高，并且不需要任何参数校准。此外，该方法扩展到粘性流的求解，并通过分析泰勒-格林，库埃特和泊瓦伊流以及盖驱动腔测试案例验证了结果。所提出的方法显示了在黎曼问题中准确性的提高，并且不需要任何参数校准。此外，该方法扩展到粘性流的求解，并通过分析泰勒-格林，库埃特和泊瓦伊流以及盖驱动腔测试案例验证了结果。所提出的方法显示了在黎曼问题中准确性的提高，并且不需要任何参数校准。此外，该方法扩展到粘性流的求解，并通过分析泰勒-格林，库埃特和泊瓦伊流以及盖驱动腔测试案例验证了结果。