Implicit immersed boundary method integrated into the Vanka ‘big box’ smoother.

Abstract

The current study introduces a novel fully coupled monolithic solver for the direct forcing immersed boundary method (IBM) in incompressible flows. The solver simultaneously integrates pressure, velocity, nonlinear convection terms, and Lagrangian forces into a unified framework, leveraging a modified big-box Vanka smoother extended with additional Lagrange multipliers arising from the IBM formulation. Central to the approach is the use of a Schur complement decomposition, which reduces the operator size by two-thirds while preserving both stability and accuracy. The solver’s monolithic structure eliminates splitting errors and artificial pressure boundary conditions, common drawbacks of segregated methods. Additionally, the developed methodology enables high CFL numbers (up to 0.5), making it particularly effective for moving boundary simulations. Verification studies cover a broad set of benchmark problems, including both stationary and moving immersed bodies across a wide range of Reynolds numbers. These tests confirm that the solver achieves computational times comparable to existing semi-implicit methods while enhancing accuracy and stability. By addressing key challenges in high-fidelity incompressible flow simulations, the proposed method offers a robust and broadly applicable monolithic solver.