Centrifugal-Acceleration Modes for Incompressible Fluid in the Leakage Annulus Between a Shrouded Pump Impeller and Its Housing

An analysis is presented for the perturbed flow in the leakage path between a shrouded-pump impeller and its housing. A bulk-flow model is used for the analysis consisting of the path-momentum, circumferential-momentum, and continuity equations. Shear stress at the impeller and housing surfaces are modeled according to Hirs’ turbulent lubrication model. The governing equations have been used earlier to examine rotordynamic reaction forces developed by lateral and axial impeller motion. A perturbation expansion of the governing equations in the eccentricity ratio yields a set of zeroth and first-order governing equations. The zeroth-order equations define the leakage rate, and the velocity and pressure distributions for a centered impeller position. The first-order equations define the perturbations in the velocity and pressure distributions due to axial or lateral motion of the impeller. Prior analyses by the author of the perturbation equation have examined the reaction forces on the shroud due to rotor motion. These analyses have produced “resonance” phenomena associated with the centrifugal-acceleration body forces in the fluid field. In the present analysis, an algorithm is developed and demonstrated for calculating the complex eigenvalues and eigenvectors associated with these resonances. First-and second-natural-frequency eigensolutions are presented for mode shapes corresponding to lateral excitation. First-natural-frequency eigensolutions are also presented for mode shapes corresponding to axial excitation.