We present a characterization of the forcing and the sub-filter scale terms produced in the volume-filtering immersed boundary (VF-IB) method by Dave et al, JCP, 2023. The process of volume-filtering produces bodyforces in the form of surface integrals to describe the boundary conditions at the interface. Furthermore, the approach also produces unclosed subfilter scale (SFS) terms. The level of contribution from SFS terms on the numerical solution depends on the filter width. In order to understand these terms better, we take a 2 dimensional, varying coefficient hyperbolic equation shown by Brady & Liverscu, JCP, 2021. This case is chosen for two reasons. First, the case involves 2 distinct regions seperated by an interface, making it an ideal case for the VF-IB method. Second, an existing analytical solution allows us to properly investigate the contribution from SFS term for varying filter sizes. The latter controls how well resolved the interface is. The smaller the filter size, the more resolved the interface will be. A thorough numerical analysis of the method is presented, as well as the effect of the SFS term on the numerical solution. In order to perform a direct comparison, the numerical solution is compared to the filtered analytical solution. Through this, we highlight three important points. First, we present a methodical approach to volume filtering a hyperbolic PDE. Second, we show that the VF-IB method exhibits second order convergence with respect to decreasing filter size (i.e. making the interface sharper). Finally, we show that the SFS term scales with square the filter size. Large filter sizes require modeling the SFS term. However, for sufficiently finer filters, the SFS term can be ignored without any significant reduction in the accuracy of solution.

翻译：本文对Dave等人于2023年在《计算物理学杂志》上提出的体积滤波浸没边界（VF-IB）方法中产生的强迫项与亚滤波尺度项进行了系统表征。体积滤波过程会生成以表面积分形式描述的界面边界条件体积力。此外，该方法还会产生未封闭的亚滤波尺度（SFS）项。SFS项对数值解的贡献程度取决于滤波宽度。为深入理解这些项的特性，我们采用Brady与Liverscu于2021年在《计算物理学杂志》中提出的二维变系数双曲型方程作为研究对象。选择该算例基于双重考量：其一，该问题包含由界面分隔的两个独立区域，是验证VF-IB方法的理想案例；其二，现有解析解使我们能系统研究不同滤波尺度下SFS项的贡献规律，其中滤波尺度直接控制界面分辨率——滤波尺度越小，界面分辨率越高。本文对该方法进行了全面的数值分析，并探讨了SFS项对数值解的影响。为进行直接对比，我们将数值解与经过滤波处理的解析解进行比较。通过系统研究，我们阐明了三个重要结论：首先，提出了一种对双曲型偏微分方程进行体积滤波的体系化方法；其次，证明了VF-IB方法在减小滤波尺度（即提升界面锐度）时具有二阶收敛特性；最后，揭示了SFS项与滤波尺度的平方成正比。大滤波尺度需要建立SFS项模型，但当滤波尺度足够精细时，忽略SFS项不会对求解精度产生显著影响。