A Finite-Element Perturbation Approach to Fluid/Rotor Interaction in Turbomachinery Elements. Part 1: Theory

The vibrational characteristics of a rotor that is in contact with a fluid in an annular clearance gap, as dictated by the fluid forces in the gap, are investigated. The “rotor” here is a general term that may refer to the shaft segment within the housing of an annular seal, on the simple end of the application spectrum, or the shroud-seal assembly in a shrouded-impeller stage of a turbomachine, on the complex end. The disturbance under consideration involves the axis of rotation, and includes a virtual lateral eccentricity, together with a whirling motion around the housing centerline. Uniqueness of the computational model stems from the manner in which the rotor eccentricity is physically perceived and subsequently incorporated. It is first established that the fluid reaction components arise from infinitesimally small deformations with varied magnitudes which are experienced by an assembly of finite elements in the rotor-to-housing gap as the gap becomes distorted due to the rotor virtual eccentricity. The idea is then cast into a perturbation model in which the perturbation equations emerge from the flow-governing equations in their discrete finite-element form as opposed to the differential form, which is traditionally the case. As a result, restrictions on the rotor-to-housing gap geometry, or the manner in which the rotor virtual eccentricity occurs are practically nonexisting. While the emphasis in this paper is on the theoretical model, a representative application of the model and assessment of the numerical results are the focus of a companion paper that is being published concurrently.