A Pressure-Robust Immersed Interface Method for Discrete Surfaces

Abstract

The immersed interface method (IIM) for fluid-structure interaction imposes discontinuities in the fluid stress along immersed boundaries that are generated by forces concentrated along those boundaries. For a viscous incompressible fluid, imposing these discontinuities requires decomposing the boundary force into its normal and tangential components, which determine jump conditions for the pressure and velocity gradient. Previously, we developed an IIM for C0 triangulated surfaces, with a focus on piecewise linear surface representations. In this setting, the normal and tangent vectors of the discrete surface are constant on each element, and that method uses those piecewise constant vectors to determine the normal and tangential force components and, ultimately, the jump condition. We demonstrated that this is substantially more accurate than immersed boundary methods that use regularized delta functions at corresponding grid resolutions for situations in which shear stresses dominate. However, this IIM formulation struggles to accurately capture pressure loads. Here, we identify that the primary cause of this limitation is the discontinuous surface normal inherent in C0 triangulated surfaces. We propose a procedure that uses approximations of the surface normals that more accurately account for the curvature and avoid discontinuities in the reconstructed normal vectors. We investigate two ways to construct the continuous surface normal approximation: a standard L2 projection of the discontinuous surface normal field into a continuous finite element space, and constructing vertex normal vectors using inverse centroid-distance weighting and applying linear interpolation across each element. Numerical experiments show that the use of jump conditions computed with reconstructed continuous normal vector fields reduce leakage by up to six orders of magnitude across a range of pressures.

Figures (16)

• Figure 1: A Lagrangian coordinate system represents represents the interface, $\Gamma_t$. At time $t$, the location of the interface in Eulerian coordinates is $\boldsymbol{\chi}(\mathbf{X},t)$.
• Figure 2: Two-dimensional schematic of a discrete immersed boundary. Normal vectors $\hat{\mathbf n}$ are discontinuous at the junctions between neighboring elements (left). Globally continuous normal vector field $\tilde{\mathbf n}$ (right).
• Figure 3: The top left, top right, and bottom panels show the vorticity field and $\varepsilon_{\mathbf{X}}$ computed using flat, inverse centroid-weighted, and projected normals with eight levels of grid refinement.
• Figure 4: Time histories of lift and drag for flow past a circular cylinder for $\ell_\text{max} = 6$, $7$, and $8$. From top to bottom, the normal vectors used are flat, inverse centroid-weighted, and projected. Both lift and drag exhibit convergence for each representation of the normal vector.
• Figure 5: The pressurized cylinder is visualized in the entire fluid domain. The shaded region inside the cylinder shows the region in which $Q$ is applied to impose $p=100$. We use $N=32$ grid cells in both directions.