A stable FSI algorithm for light rigid bodies in compressible flow

In this article we describe a stable partitioned algorithm that overcomes the added mass instability arising in fluid–structure interactions of light rigid bodies and inviscid compressible flow. The new algorithm is stable even for bodies with zero mass and zero moments of inertia. The approach is based on a local characteristic projection of the force on the rigid body and is a natural extension of the recently developed algorithm for coupling compressible flow and deformable bodies [1], [2] and [3]. The new algorithm advances the solution in the fluid domain with a standard upwind scheme and explicit time-stepping. The Newton–Euler system of ordinary differential equations governing the motion of the rigid body is augmented by added mass correction terms. This system, which is very stiff for light bodies, is solved with an A-stable diagonally implicit Runge–Kutta scheme. The implicit system (there is one independent system for each body) consists of only 3d+d23d+d2 scalar unknowns in d=2d=2 or d=3d=3 space dimensions and is fast to solve. The overall cost of the scheme is thus dominated by the cost of the explicit fluid solver. Normal mode analysis is used to prove the stability of the approximation for a one-dimensional model problem and numerical computations confirm these results. In multiple space dimensions the approach naturally reveals the form of the added mass tensors in the equations governing the motion of the rigid body. These tensors, which depend on certain surface integrals of the fluid impedance, couple the translational and angular velocities of the body. Numerical results in two space dimensions, based on the use of moving overlapping grids and adaptive mesh refinement, demonstrate the behavior and efficacy of the new scheme. These results include the simulation of the difficult problems of shock impingement on an ellipse and a more complex body with appendages, both with zero mass.