In this study, the nodal discontinuous Galerkin method is formulated in three-dimensions and applied to simulate three-dimensional non-cavitating/cavitating flows. For this aim, the three-dimensional preconditioned Navier-Stokes equations based on the artificial compressibility approach considering appropriate source terms to model cavitating phenomena are used. The spatial derivative terms in the resulting equations are discretized by utilizing the nodal discontinuous Galerkin method on tetrahedral elements and the derivative of the solution vector with respect to the artificial time is discretized by applying an explicit time integration method. An artificial viscosity method is formulated in three-dimensions and applied to properly capture the discontinuities in the cavitating flow-field. Here, different non-cavitating/cavitating flow problems including non-cavitating flow in a 3:1:1 driven cavity, non-cavitating flow between two concentric rotating spheres, cavitating flow over a sphere and cavitating flow over a wavy cylinder are solved to examine the accuracy and robustness of the proposed solution method. These flow problems are selected to properly assess the numerical method in the presence of straight and curved boundaries and for different Reynolds and cavitation numbers. From linear to cubic polynomial degrees are used in the simulations and the numerical results are compared with the available analytical and numerical results. Indications are that the present numerical method based on the NDGM formulated in three-dimensions can accurately simulate the three-dimensional non-cavitating/cavitating flows.

在这项研究中，节点不连续伽辽金方法在三维中被制定并应用于模拟三维非空化/空化流动。为此，使用基于人工可压缩性方法的三维预处理 Navier-Stokes 方程，考虑适当的源项来模拟空化现象。所得方程中的空间导数项通过利用四面体单元上的节点不连续伽辽金方法进行离散化，并通过应用显式时间积分方法将解向量相对于人工时间的导数离散化。人工粘度方法在三个维度中制定并应用于正确捕获空化流场中的不连续性。这里，解决了不同的非空化/空化流动问题，包括 3:1:1 驱动腔中的非空化流动、两个同心旋转球体之间的非空化流动、球体上的空化流动和波状圆柱体上的空化流动，以检查所提出的求解方法的准确性和鲁棒性。选择这些流动问题是为了在存在直线和曲线边界以及不同雷诺数和空化数的情况下正确评估数值方法。模拟中使用从线性到三次多项式的次数，并将数值结果与可用的解析和数值结果进行比较。表明现有的基于三维建模的NDGM的数值方法可以准确地模拟三维非空化/空化流动。