In previous papers (Axisa & Antunes 1992, Antunes et al. 1996) the authors presented simplified theoretical models for both centered and eccentric immersed rotors, under moderate confinement (with reduced gaps δ = H/R of about 0.1, where H is the annular fluid gap and R is the rotor radius). In those papers rotor dynamics were modeled using linearised flow equations on the gap-averaged fluctuating quantities. Such simplified models enabled us to predict with reasonable success the modal parameters of the coupled system and the stability boundaries, as a function of the annulus geometry and rotor spinning velocity.
However, experiments have shown a progressive deterioration of the theoretical predictions at high spinning velocities as well as at large eccentricities. Such deterioration might be due to nonlinear effects which became significant near and beyond the stability boundaries. Hence, better predictions might be produced if the nonlinear terms of the flow equations were fully accounted — an issue which is pursued in this work. Because such nonlinear analysis is quite involved, this paper will focus on the simpler case of planar motions, in order to emphasize the main aspects of our approach.
A direct integration of the continuity and momentum equations leads to extremely lengthy formulations, which are very difficult to manipulate — even using symbolic computational tools. In this paper we discuss several strategies to obtain approximate solutions of the flow equations. Then we proceed with the actual approach used, which consists on the exact integration of the continuity equation and an approximate solution of the momentum equation, based on a Fourier representation of the azimuthal pressure gradient. We then obtain an exact formulation for the dynamic flow force.
The solution thus obtained is discussed in connection with physical phenomena. Numerical simulations of the nonlinear system dynamics are presented, which display interesting features. The linearised and the fully nonlinear models produces very similar results when the eccentricity and the spinning velocity are low. However, if such conditions are not met, the qualitative behavior stemming from these models is quite distinct. Experimental results indicate that the nonlinear flow model leads to better predictions of the rotor dynamics when the eccentricity is significant, when approaching instability and for linearly unstable regimes.