An implicit method for coupled flow–body dynamics

We propose an efficient method for computing coupled flow–body dynamics. The time-stepping is implicit, and uses an iterative method (preconditioned GMRES) to solve the flow–body equations. The preconditioner solves a decoupled version of the equations which involves only the inversion of banded matrices, and requires a small number of iterations per time step. We use the method to probe the instability to horizontal motions of an elliptical body with simple vertical motions: flapping and rising. In both cases a linear instability to horizontal motion sets in above a critical Reynolds number, leading to a stable oscillatory state. The pressure forces play a destabilizing role against the stabilizing viscous forces, with oscillatory time scales set by either external flapping or the intrinsic flow–body coupling. The latter lowers the instability threshold in Reynolds number.