This paper presents an immersed boundary method for weak enforcement of Dirichlet boundary conditions on surfaces that are immersed in the stationary background discretizations. An interface stabilized form is developed by applying the Variational Multiscale Discontinuous Galerkin (VMDG) method at the immersed boundaries. The formulation is augmented with a variationally derived ghost-penalty type term. The weak form of the momentum balance equations is embedded with a residual-based turbulence model for incompressible turbulent flows. A significant contribution in this work is the variationally derived analytical expression of the Lagrange multiplier for weak enforcement of the Dirichlet boundary conditions at the immersed boundary. In addition, the analytical expression for the interfacial stabilization tensor emerges which accounts for the geometric aspects of the cut elements that are produced when the immersed surface geometry traverses the underlying mesh. A unique attribute of the fine-scale variational equation is that it also yields a posteriori error estimator that can evaluate the local error in weak enforcement of the essential boundary conditions at the embedded boundaries. The method is shown to work with meshes comprised of hexahedral and tetrahedral elements. Numerical experiments show that the norm of the stabilization tensor varies spatially and temporally as a function of the flow physics at the embedded boundary. Test cases with increasing levels of complexity are presented to validate the method on benchmark problems of flows around cylindrical and spherical geometric shapes, and turbulent features of the flows are analyzed.

本文提出了一种浸入式边界方法，用于在浸入静止背景离散化的表面上弱执行 Dirichlet 边界条件。通过在浸没边界处应用变分多尺度不连续 Galerkin (VMDG) 方法开发了界面稳定形式。该公式增加了一个变分派生的幽灵惩罚类型术语。动量平衡方程的弱形式嵌入了不可压缩湍流的基于残差的湍流模型。这项工作的一个重要贡献是拉格朗日乘数的变分解析表达式，用于在浸没边界处弱执行狄利克雷边界条件。此外，界面稳定张量的解析表达式出现，它解释了当浸没的表面几何形状穿过底层网格时产生的切割元素的几何方面。精细变分方程的一个独特属性是它还产生一个后验误差估计器，该估计器可以评估嵌入边界处基本边界条件的弱执行时的局部误差。该方法显示适用于由六面体和四面体单元组成的网格。数值实验表明，稳定张量的范数在空间和时间上随嵌入边界处的流动物理变化而变化。