Characterization of the forcing and sub-filter scale terms in the volume-filtering immersed boundary method

Abstract

We present a characterization of the forcing and sub-filter scale terms produced in the volume-filtering immersed boundary (VF-IB) method by Dave et al. [5]. 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 terms called ${\tau }_{\mathrm{sfs}}$. The level of contribution from ${\tau }_{\mathrm{sfs}}$ on the numerical solution depends on the filter width ${\delta }_f$. In order to understand these terms better we take a 2 dimensional, varying coefficient hyperbolic equation shown by Brady and Livescu [3]. This case is chosen for two reasons. First, the case involves 2 distinct regions separated 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 ${\tau }_{\mathrm{sfs}}$ for varying ${\delta }_f$. The filter width controls how well resolved the interface is. The smaller the filter width, the more resolved the interface will be. A thorough numerical analysis of the method is presented, as well as the effect of ${\tau }_{\mathrm{sfs}}$ 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 ${\delta }_f$ (i.e. making the interface sharper). Finally, we show that ${\tau }_{\mathrm{sfs}}$ scales with ${\delta }_f^2$. Large filter widths would require a modeling approach to sufficiently resolve ${\tau }_{\mathrm{sfs}}$. However for finer filter widths that have a sufficiently sharp interface, ${\tau }_{\mathrm{sfs}}$ can be ignored without any significant reduction in the accuracy of solution. We show that through the inclusion of these unclosed terms, the VF-IB method can bridge the gap between fully modeled and fully resolved methods by providing accurate results when the filter width is of the same order as the characteristic solid corrugation length scale.