An Immersed Boundary Method for Rigid Bodies

TL;DR

This paper introduces an immersed boundary method for simulating flows around rigid bodies, ensuring exact no-slip constraints and efficient solutions across various Reynolds numbers, including steady Stokes flow.

Contribution

It presents a novel IB approach that enforces no-slip constraints exactly without penalty methods, using an efficient preconditioner and demonstrating robustness and accuracy across different flow regimes.

Findings

Achieves exact no-slip boundary conditions in IB simulations.

Requires only a few multigrid cycles for efficient preconditioning.

Demonstrates first-order convergence to analytical and benchmark solutions.

Abstract

We develop an immersed boundary (IB) method for modeling flows around fixed or moving rigid bodies that is suitable for a broad range of Reynolds numbers, including steady Stokes flow. The spatio-temporal discretization of the fluid equations is based on a standard staggered-grid approach. Fluid-body interaction is handled using Peskin's IB method; however, unlike existing IB approaches to such problems, we do not rely on penalty or fractional-step formulations. Instead, we use an unsplit scheme that ensures the no-slip constraint is enforced exactly in terms of the Lagrangian velocity field evaluated at the IB markers. Fractional-step approaches, by contrast, can impose such constraints only approximately. Imposing these constraints exactly requires the solution of a large linear system that includes the fluid velocity and pressure as well as Lagrange multiplier forces that impose the motion of the body. To solve this system efficiently, we develop a preconditioner for the constrained IB formulation that is based on an analytical approximation to the Schur complement. This approach is enabled by the near translational and rotational invariance of Peskin's IB method. We demonstrate that only a few cycles of a geometric multigrid method for the fluid equations are required in each application of the preconditioner, and we demonstrate robust convergence of the overall Krylov solver despite the approximations made in the preconditioner. We apply the method to a number of test problems at zero and finite Reynolds numbers, and we demonstrate first-order convergence of the method to several analytical solutions and benchmark computations.