A THEORETICAL MODEL FOR NONLINEAR PLANAR MOTIONS OF ROTORS UNDER FLUID CONFINEMENT

Abstract

Following previous papers by Axisa, Antunes and co-workers, the authors address a theoretical model for immersed rotors, under moderate confinement, using simplified flow equations on the gap-averaged fluctuating quantities. However, in contrast to our previous efforts, the nonlinear terms of the flow equations are here fully accounted. 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. Here, in order to solve the flow equations, we perform an exact integration of the continuity equation and an approximate solution of the momentum equation, based on a Fourier representation of the azimuthal pressure gradient. Then, an exact formulation for the dynamic flow force can be obtained. Our solution is discussed in connection with physical phenomena. Numerical simulations of the nonlinear rotor-flow coupled system are presented, showing that the linearized and the fully nonlinear models produces similar results when the eccentricity and the spinning velocity are low. However, if such conditions are not met, the qualitative dynamics stemming from these models are 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.