An immersed boundary method for rigid bodies

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. Fractionalstep approaches, by contrast, can impose such constraints only approximately, which can lead to penetration of the flow into the body, and are inconsistent for steady Stokes flow. Imposing no-slip 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. The principal contribution of this paper is that it develops an efficient preconditioner for this exactly constrained IB formulation which 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 empirically observe that to control the condition number of the coupled linear system while also keeping the rigid structure impermeable to fluid, we need to place the immersed boundary markers at a distance of about two grid spacings, which is significantly larger from what has been recommended in the literature for elastic bodies. We demonstrate the advantage of our monolithic solver over split solvers by computing the steady state flow through a two-dimensional nozzle at several Reynolds numbers. We apply the method to a number of benchmark problems at zero and finite Reynolds numbers, and we demonstrate first-order convergence of the method to several analytical solutions and benchmark computations.