Stability of nonlinear solutions of rigid body-viscous flow interaction

The finite-element method is extended to cover interactions between viscous flow and a moving body and to investigate the stability of the resulting flow. Special consideration is given to the modeling of a two-dimensional incompressible flow over a solid body elastically supported or alternatively undergoing a specified harmonic oscillation, as well as to a particular problem of a body of arbitrary shape undergoing simple harmonic motion in otherwise undisturbed fluid. The finite element modeling is based on a velocity-pressure primitive variable representation of the Navier-Stokes equations, using curved isoparametric elements with quadratic interpolation for the velocities and bilinear for pressure. The nonlinear boundary conditions on the moving body are represented to the first order in the body amplitude parameter, using the method of averaging to analyze the resulting periodic motion of the fluid. Small perturbations are introduced in this motion to examine the stability of the solution, applying the Floquet theory to examine the nature of the small perturbations.