ROTORDYNAMIC FLUID FORCES ON WHIRLING AND CAVITATING RADIAL IMPELLERS

This paper investigates the linearized dynamics of the rotordynamic forces exerted by the fluid on the rotor in whirling and cavitating radial impellers with thin logarithmic blades and constant eccentricity and whirl speed. The flow is modeled as an incompressible, inviscid, fully-guided liquid except on the suction sides of the blades, where attached cavitation occurs in a small layer of given acoustic admittance depending on the assumed values of the layer thickness and void fraction. Constant boundary conditions for the total pressure are imposed at the inlet and outlet sections. The three-dimensional unsteady governing equations are written in rotating orthonormal logarithmic spiral coordinates, linearized for small-amplitude whirl perturbations of the mean steady flow, and solved by modal decomposition. Rotordynamic fluid forces in centrifugal pumps are found to be almost insensitive to cavitation; also, they do not undergo the internal flow resonances in the blade channels predicted by similar flow models and observed in whirling and cavitating axial inducers. Comparison with the available experimental results shows that the present theory underestimates the intensity of rotordynamic impeller forces, but correctly captures their observed parabolic trend as functions of the whirl frequency, thus indicating that it can usefully contribute to identify the main physical phenomena involved and provide useful practical indications on their dependence on the relevant flow conditions and parameters. INTRODUCTION Local flow phenomena like tip leakage capable of interfering with the fluid dynamic loads on the blades are known to be the dominant source of rotordynamic whirl forces in compressible flow machines (Thomas, 1958 [1], Alford, 1958 [2], Martinez-Sanchez et al., 1995 [3], Song and MartinezSanchez, 1997a [4], Martinez-Sanchez, 1997b [5]). On the other hand, in turbopumps also the reaction of the entire flow to the impeller whirl motion can often be significant (d'Auria, d’Agostino and Brennen, 1995 [6]; d’Agostino and d’Auria, 1997 [ 7 ]; d’Agostino, d’Auria and Brennen, 1998 [ 8 ], d’Agostino and Venturini, 2002 [ 9 ]). Previous turbopump research has mainly focused on the origin and analysis of rotordynamic impeller forces in noncavitating conditions (Chamieh et al., 1985 [10]; Jery, Acosta and Caughey, 1985 [11]; Jery, 1987 [12]; Shoji and Ohashi, 1987 [13]; Ohashi and Shoji, 1987 [14]; Adkins and Brennen, 1988 [15]; Arndt et al., 1989 [16]; Arndt et al., 1990 [17]; Tsujimoto et al., 1997 [18]; Uy and Brennen, 1999 [19]; Baskharone, 1999 [20], Hiwata and Tsujimoto, 2002 [21]). Because of their greater complexity, rotordynamic fluid forces in whirling and cavitating turbopump impellers have so far received comparatively less attention in the open literature and a satisfactory understanding of their behavior is still lacking. However, it is widely recognized that turbopump cavitation in axial inducers can promote the onset of dangerous self-sustained whirl instabilities (Rosenmann, 1965 [22]) and substantially alters the behavior of fluid-induced rotordynamic forces (Arndt and Franz, 1986 [ 23 ]; Brennen, 1994 [ 24 ]; Bhattacharyya, 1994 [25]). The available experimental evidence indicates that inducer cavitation reduces the magnitude of the rotordynamic fluid forces, significantly affecting the added mass of the rotor. The consequent increase of the critical speeds is of special relevance to highly-loaded supercritical machines, as commonly used in liquid propellant rocket feed systems. A second major effect of inducer cavitation is the introduction of a complex oscillatory dependence of the rotordynamic fluid forces on the whirl frequency. This finding seems to indicate the occurrence of resonance phenomena in the compressible cavitating flow inside the blade channels under the excitation imposed by the eccentric motion of the rotor. Earlier theoretical analyses confirmed the presence of internal flow resonances and indicate that bubble dynamic effects do not play a major role, except, perhaps, at extremely high whirl speeds (d'Auria, d’Agostino and Brennen, 1995 [26]; d’Agostino and d’Auria, 1997 [ 27 ]; d’Agostino, d’Auria and Brennen, 1998 [ 28 ], d’Agostino and Venturini, 2002 [9]). On the other hand, no resonant phenomena seem to occur radial impellers, where the limited available evidence indicates that cavitation only has a marginal effect on rotordynamic whirl forces (Franz et al., 1989 [29]). Following up on our work on whirling and cavitating inducers, we now apply a similar approach to the analysis of the unsteady two-dimensional flow in whirling radial impellers with