Immersed boundary method with improved implicit direct-forcing for fluid–structure interaction problems

Abstract

An improved implicit direct-forcing immersed boundary method (DF-IBM) is proposed for simulating interactions between incompressible fluid flows and complex rigid structures undergoing arbitrary free motion, commonly referred to as fluid–rigid body interaction problems. The proposed approach harnesses the pressure implicit with splitting of operators (PISO) algorithm to efficiently handle the dual constraints of the fluid–solid system in a segregated manner. Consequently, the divergence-free condition is maintained throughout the Eulerian domain, while the kinematic no-slip velocity boundary condition is exactly enforced on the immersed boundary, also termed as the fluid–structure interface. A new pressure Poisson equation (PPE) is derived, incorporating the boundary force directly where the no-slip condition is satisfied. This approach avoids altering the coefficient matrix of the PPE, which could otherwise introduce convergence issues, enabling the use of fast iterative PPE solvers without modifications. The improvement involves integrating Lagrangian weight methods, having better reciprocity over the IBM-related linear operators, within the implicit formulation. An additional force initialization scheme is introduced to accelerate the convergence of the no-slip boundary condition, thereby improving the algorithm’s performance. The Navier-Stokes equations are coupled with the rigid body dynamics, described by the Newton-Euler equations, within the improved DF-IBM framework. Both explicit and implicit coupling algorithms are developed to address weakly and strongly coupled fluid–rigid body interaction problems, respectively, under a partitioned approach. Stability and convergence issues, particularly stemming from critical solid–fluid density ratios and/or the rigid body approximation of the internal mass effects (IME) in rotational dynamics, are mitigated using a fixed relaxation technique for the rigid body kinematics. For implicit coupling, a fixed-point strategy is employed, complemented by the relaxation technique used for the IME to ensure robustness. Additionally, the proposed coupling algorithms leverage the DF-IBM formulation and the predictor-corrector strategy of the PISO solution algorithm, by excluding the momentum predictor step and the time-intensive corrector loops from the implicit iterations. The proposed method is validated through various stationary, prescribed, and freely moving immersed boundary cases, with results compared against experimental and numerical data from the literature. The method demonstrates robustness, accuracy, and efficiency in handling the complex dynamics of fluid–rigid body interactions across a range of challenging scenarios. The suggested improvements integrate seamlessly into existing incompressible fluid solvers with minimal adjustments to the original system of equations, highlighting their ease of implementation. Finally, the present work is implemented within the cell-centred finite volume approach inside the open-source C++ toolbox OpenFOAM environment, version 7.0 of the OpenFOAM Foundation variant.PhD in Energy and Powe