This paper presents a new stabilized form of incompressible Navier–Stokes equations for weak enforcement of Dirichlet boundary conditions at immersed boundaries. The boundary terms are derived via the Variational Multiscale (VMS) method which involves solving the fine-scale variational problem locally within a narrow band along the boundary. The fine-scale model is then variationally embedded into the coarse-scale form that yields a stabilized method which is free of user defined parameters. The derived boundary terms weakly enforce the Dirichlet boundary conditions along the immersed boundaries that may not align with the inter-element edges in the mesh. A unique feature of this rigorous derivation is that the structure of the stabilization tensor which emerges is naturally endowed with the mathematical attributes of area-averaging and stress-averaging. The method is implemented using 4-node quadrilateral and 8-node hexahedral elements. A set of 2D and 3D benchmark problems is presented that investigate the mathematical attributes of the method. These test cases show that the proposed method is mathematically robust as well as computationally stable and accurate for modeling boundary layers around immersed objects in the fluid domain.

中文翻译：

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不可压缩流浸入边界条件的变分多尺度方法

本文提出了一种新的稳定形式的不可压缩纳维-斯托克斯方程，用于在浸没边界弱执行狄利克雷边界条件。边界项是通过多尺度变分 (VMS) 方法导出的，该方法涉及在沿边界的窄带内局部求解精细尺度变分问题。然后，精细尺度模型被可变地嵌入到粗尺度形式中，从而产生不受用户定义参数影响的稳定方法。导出的边界项沿浸入边界弱强制狄利克雷边界条件，这些边界可能与网格中的单元间边缘不对齐。这种严格推导的一个独特之处在于，所出现的稳定张量的结构自然地被赋予了面积平均和应力平均的数学属性。该方法使用 4 节点四边形和 8 节点六面体单元来实现。提出了一组 2D 和 3D 基准问题来研究该方法的数学属性。这些测试案例表明，所提出的方法在数学上是稳健的，并且在计算上稳定且准确，可以对流体域中浸入物体周围的边界层进行建模。