An Immersed Interface Method for Incompressible Flows and Near Contact

Abstract

We present an enhanced immersed interface method for simulating incompressible fluid flows in thin gaps between closely spaced immersed boundaries. This regime, common in engineered structures such as including tribological interfaces and bearing assemblies, poses significant computational challenges because of limitations in grid resolution and the prohibitive cost of mesh refinement near contact. The immersed interface method imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Our approach introduces a bilinear velocity interpolation operator that incorporates jump conditions from multiple nearby interfaces when they occupy the same interpolation stencil. Numerical results demonstrate substantial improvements in both interface and Eulerian velocity accuracy compared to lubrication-based immersed boundary and immersed interface methods. The proposed method improves upon previous interpolation schemes, and eliminates the need for prior knowledge of interface orientation or geometry. This makes it broadly applicable to a wide range of fluid--structure interaction problems involving near-contact dynamics.

Figures (16)

• Figure 1: A Lagrangian coordinate system represents represents the interface, $\Gamma_t$. At time $t$, the position of the interface in Eulerian coordinates is $\boldsymbol{\chi}(\mathbf{X},t)$.
• Figure 2: Velocity interpolation from Cartesian grid velocities (blue) to quadrature point $\boldsymbol{\alpha}_k$ (red) on $\Gamma_t^1$. Velocities at $x_1$, $x_2$, and $x_3$ require correction terms to account for the discontinuity in the velocity gradient generated by the interface.
• Figure 3: Schematic for velocity interpolation when two interfaces lie within the same bounding box. Weights $\xi$ and $\zeta$ are the $x$ and $y$ components of the unsigned distance from $\tilde{\mathbf{x}}_2$ to $\mathbf{x}_2$.
• Figure 4: Velocity interpolation schematics are shown for various orientations of $\Gamma_t^1$ and $\Gamma_t^2$ when both interfaces lie within one grid cell. The intersection of ray $\mathbf{r}$ with $\Gamma_t^2$ is denoted by $\tilde{x}$. Panels (a)-(c) show the possible orientations when $\Gamma_t^1$ and $\Gamma_t^2$ each isolate one node. Panels (d) and (e) show the possible orientations when $\Gamma_t^1$ isolates one node and $\Gamma_t^2$ isolates two nodes. Panel (f) shows the orientation in which both $\Gamma_t^1$ and $\Gamma_t^2$ isolate two nodes. Panel (g) shows the orientation in which $\Gamma_t^1$ isolates one node and $\Gamma_t^2$ isolates three nodes. All other possible orientations can be achieved by rotating both interfaces and swapping $\Gamma_t^1$ with $\Gamma_t^2$.
• Figure 5: Shearing parallel plate schematic. Both plates move with equal and opposite velocity.