A semi-implicit semi-Lagrangian method for simulating immersed boundary motion under high inertia and elasticity

In this paper, we present an efficient and stable fractional-step 2D immersed boundary (IB) method for solving the interaction problems between bulk fluid and elastic interface, in particular, when the fluid inertia and the interfacial elasticity are the significant factors affecting its dynamics. In myriads of real-world applications, the effects of high inertia and elasticity are dominant. So the complex fluid dynamics under such harsh conditions is an important topic in computational physics and is inherently challenging due to high computational complexity. In turn, it requires to solve the governing equations of elastic interfacial motion in an implicit manner so that more stable simulations can be performed by relaxing the Courant-Friedrichs-Lewy (CFL) condition. The contributions of the proposed approach are three folds. First, an iteration-free semi-Lagrangian method is employed in Navier-Stokes (NS) equations. Second, the elastic force acting along the interface is treated semi-implicitly in IB formulations. Both approaches improve the numerical stability associated with the high fluidic inertia and interfacial elasticity. Finally, to solve the resulting linear system, our novel idea is to transform the original 3-by-3 block matrix system into a reduced 2-by-2 block matrix system using a discrete projection operator in a staggered grid, and then explicitly represent the exact solution via the Schur complement of the Helmholtz operator. Owing to this feature, we refer to this proposed approach as reduced immersed boundary method (rIBM). We show that the two systems are equivalent in theory, whereas the conventional immersed boundary projection method (IBPM) modifies the discrete momentum equation in the original system. A series of numerical tests is conducted to confirm the stability of the rIBM using relatively larger time-step sizes, specifically with Reynolds number and inverse capillary number equal to or larger than approximately 1000. By estimating the computational time, the numerical efficiency of the proposed method is further verified in comparison with the conventional IBPM and the Crank-Nicolson scheme-based IB method. In conclusion, the proposed approach not only improves the numerical stability, but also increases the computational speed, suitable for solving more realistic problems.