## Key

1414

## Conference Year

2014

## Keywords

centrifugal compressor, impeller wake vane interaction, radial waves, pressure waves, aeromechnics

## Abstract

Traditionally, industrial centrifugal compressors have had fewer aeromechanical issues than axial compressors. However, in the development of the next generation high power density industrial compressors, aeromechanical issues are of increasing concern. In particular, a vaned diffuser is frequently being used to increase the compressor efficiency. The resulting impingement of the impeller wakes on the downstream diffuser vanes generates a series of strong acoustic waves. Both experiments [1] and simulations [2] have shown that under certain operating conditions, these acoustic waves generated by the impeller-diffuser vane interactions are strong enough to cause impeller blade failures. To analyze this aeromechanical risk, three primary physical processes are involved: (1) the impeller wakes traveling downstream; (2) the wake interaction with the diffuser vane generating the pressure waves; (3) the pressure waves traveling upstream to excite the impeller. This paper focuses on the development of a mathematical model for the wake and pressure wave propagation in a radial duct with a mean swirling flow, processes 1 and 3. These wave propagation properties are also fundamental to the linearized unsteady aerodynamic analysis of thin airfoil radial cascades required to model the wake-vane interaction, process 2. In subsonic centrifugal compressors, the vaneless space between the impeller and the vaned diffuser is usually in the shape of a thin annulus. To reduce the complexity of the problem, the flow is assumed to be inviscid and two dimensional in the radial and circumferential directions under an isentropic process with no heat transfer. The unsteady flow is modeled as a small perturbation superimposed upon the steady mean flow. Assuming a low Mach number, the mean flow field is found to be irrotational. As shown by Goldstein [3], in a 2-D irrotational mean flow, the pressure wave and vorticity wave, i.e. the wake, are uncoupled. Thus the linearized unsteady Euler equations can be split into two sets of governing equations, one for the pressure waves and the other one for the vorticity waves. To obtain an analytical solution, perturbations are assumed to be harmonic in time and in the circumferential direction, thereby transforming the partial differential governing equations into ordinary differential equations. By applying the boundary condition that the pressure perturbation goes to zero in the limit where the radius is infinite, the perturbation potential associated with the pressure wave is found to be a combination of Hankel functions of the first and second kind. The governing equations for the vorticity wave suggest that the solutions contain both the vorticity wave and an induced hydrodynamic pressure wave due to the non-uniform swirling mean flow. The circumferential perturbation velocity of the vorticity wave is found to decrease with radius whereas the radial perturbation velocity of the vorticity wave increases with radius. The diverging behavior is explained in a similar way as Rayleigh’s criterion for inviscid instability of a basic swirling flow.

*Analytical Modeling of the Unsteady Wave Propagation in the Vaneless Space of a Low Speed Centrifugal Compressor*