A stable fluid–structure-interaction solver for low-density rigid bodies using the immersed boundary projection method

Abstract

Dispersion of low-density rigid particles with complex geometries is ubiquitous in both natural and industrial environments. We show that while explicit methods for coupling the incompressible Navier-Stokes equations and Newton's equations of motion are often sufficient to solve for the motion of cylindrical particles with low density ratios, for more complex particles -such as a body with a protrusion -they become unstable. We present an implicit formulation of the coupling between rigid body dynamics and fluid dynamics within the framework of the immersed boundary projection method. Similarly to previous work on this method, the resulting matrix equation in the present approach is solved using a block-LU decomposition. Each step of the block-LU decomposition is modified to incorporate the rigid body dynamics. We show that our method achieves second-order accuracy in space and first-order in time (third-order for practical settings), only with a small additional computational cost to the original method. Our implicit coupling yields stable solution for density ratios as low as 10 −4 . We also consider the influence of fictitious fluid located inside the rigid bodies on the accuracy and stability of our method.

A stable partitioned FSI algorithm for rigid bodies and incompressible flow. Part I: Model problem analysis

“…While this approach is common to both algorithms, there are important differences. The principal difference is that the AMP-RB algorithm uses the interface conditions in (16) and (17) for addedmass, and the condition in (22) for added-damping in the stage involving the elliptic problem for the fluid pressure. Thus, the pressure and the acceleration of the rigid body are determined together in one stage of the AMP-RB algorithm, while the velocity of the body is advanced in time in a subsequent stage.…”

“…FSI problems of this type occur in a wide variety of applications, such as ones involving particulate flows (suspensions, sedimentation, fluidized beds), valves and moving appendages, buoy structures, ship maneuvering, and underwater vehicles, to name a few. A wide range of numerical techniques have been developed to simulate such FSI problems, including arbitrary Lagrangian-Eulerian (ALE) methods [1][2][3], methods based on level-sets [4,5], fictitious domain methods [6][7][8], embedded boundary methods [9] and immersed boundary methods [10][11][12][13][14][15][16][17][18][19][20][21][22].…”

“…Many partitioned schemes for FSI problems coupling rigid bodies and incompressible flow suffer from numerical instabilities for light bodies. These instabilities usually originate from added-mass and added-damping effects as noted in [2,11,15,16,19,22,23] for example. 5 The dominant instability is often due to added-mass effects that arise from the pressure-induced force and torque on the body.…”

“…They predicated the velocity field in a temporary non-inertial frame that enabled an implicit direct solution of the rigid-body motion and an exact match of the velocity of the fluid and rigid body at the interface. Lacis et al [22] recently proposed a different non-iterative immersed boundary method that is stable for a density ratio as low as 10 −4 . Their approach is similar to the block-factorization approach used for similar FSI problems by Fedkiw et al [24] and Badia et al [25].…”

A stable partitioned FSI algorithm for rigid bodies and incompressible flow in three dimensions

“…It is worth noting that the FSI regime considered in this paper has drawn significant attention due to its importance for many applications. Besides the moving overlapping grid approach employed here, a variety of classical numerical techniques for moving complex geometries have been extended to this regime, including arbitrary Lagrangian-Eulerian (ALE) methods [16][17][18], level-set methods [19,20], fictitious domain methods [21], embedded boundary methods [22] and immersed boundary methods [11,[23][24][25][26][27][28][29][30][31][32][33][34]. Recently, new approaches to handle moving geometries have also been developed for this regime, such as methods based on boundary-integral equations [35] and implicit mesh discontinuous Galerkin methods [36,37].…”