INTERACTION OF ACOUSTIC AND VORTICAL DISTURBANCES WITH AN ANNULAR CASCADE IN A SWIRLING FLOW

H. M. Atassi,[1] A. A. Ali,

University of Notre Dame, Notre Dame, IN 46556, USA

O. V. Atassi

Pratt & Whitney, East hartford, CT 06108, USA

Abstract A linearized Euler formulation is developed for the interaction and scattering

of incident acoustic and vortical disturbances by an unloaded annular cascade in a swirling fbw. Exact nonreffecting boundary conditions for the scattered sound are applied at the inlet/outlet of the computational domain. An explicit primitive variable scheme is implemented and validated by comparison with the uniform few and narrow annulus limits. Acoustic and aerodynamic results are presented and show that the swirl changes the amplitude and radial phase of the incident disturbances and modifies the number and radial shape of the acoustic modes in the duct. These changes have significant effects on the aerodynamic and acoustic response of the cascade.

1. Introduction

For typical fan engines, the interaction of the rotor wakes with downstream guide vanes is a major source of noise and vibration. Downstream of the rotor the ft>w has a swirling motion produced by the work done on the ft>w by the rotor. The swirl strongly distorts the rotor wakes [Podboy et al., 2002] and pro­duces centrifugal forces that defects the Arid motion and couple the acoustic, entropic and vortical modes in the duct [Kerrobrock, 1977]. The present pa­per develops a model for the scattering of incident acoustic and vortical waves

by an unloaded annular cascade and examines the effect of mean fbw swirl, cascade geometry, and incident disturbances on the cascade acoustic and aero­dynamic response.

The early model for the cascade scattering problem is the two-dimensional cascade which is obtained by unrolling the annular cascade into an infinite lin­ear cascade. This approximation is valid in the limit where the annulus gap between the tip and hub radii is small compared to the average radius. For a fkt plate linear cascade in a uniform ft>w, the unsteady velocity field can be split into purely convected incident vortical disturbances and scattered poten­tial disturbances. These disturbances are only coupled by the impermeability condition along the blade surface. The linearized governing equations of the scattered field have constant coefficients and thus the problem can be formu­lated in terms of a singular integral equation [Kaji and Okazaki, 1970b, Gold­stein, 1976, Atassi, 1994]. Numerical solutions have been obtained for the un­steady pressure distribution on the blades [Smith, 1971, Ventres, 1980], and the acoustic radiation upstream and downstream [Kaji and Okazaki, 1970a, Hamad and Atassi, 1981]. More recently, analytical formulations using the Wiener – Hopf method were developed for the linear cascade aerodynamic and acoustic fields [Peake and Kerschen, 1995, Glegg, 1999].

The three dimensional geometry of the annular duct was first considered by Schulten, 1982 and Namba, 1987 for a zero stagger cascade using a singularity method. Their model accounts correctly for the duct acoustic modes. However, such methods cannot be extended to staggered and/or loaded cascades since they cannot account for the effects of nonuniform mean flow.

A linearized Euler analysis was developed by Montgomery and Verdon, 1997. At the inlet, they assumed that the gust was convected by the mean flow, thus neglecting changes in amplitude and phase of the incoming distur­bance caused by the mean flow. Golubev and Atassi, 2000b and Elhadidi et al., 2000 showed that these changes significantly modified the evolution of the unsteady incident vortical disturbances. More recently, Podboy et al., 2002 carried out measurements that show significant effect of the swirl on wake evolution. Golubev and Atassi, 2000a developed a model for the interaction of unsteady incident disturbances in a swirling mean motion with an annular cascade of unloaded blades. Numerical solutions were obtained only for the unsteady blade pressure. More recently, Atassi et al., 2004 developed a more complete analytical and numerical analysis of the cascade scattering problem in a swirling flow which gives the upstream and downstream scattered sound. They also examined the conditions that result in strong scattering.

In the present paper, the analytical and numerical analyses developed for the interaction of incident disturbances propagating in a mean swirling flow with an unloaded annular cascade are summarized, and aerodynamic and acoustic results are presented to examine the effect of cascade geometry and swirl dis­
tribution on the scattering of incident vortical and acoustic waves. The paper will also provide an accurate and efficient numerical scheme for the calculation of blade unsteady forces and the upstream and downstream radiated sound.

As a first step in the problem formulation, it is shown that the upstream disturbances cannot be specified arbitrarily. This is a result of the fact that the upstream fbw is not uniform and for subsonic fbws some information has to come from the solution inside the computational domain. The representation of upstream disturbances will be based on results from the normal mode analysis [Golubev and Atassi, 1998, Ali et al., 2000, Ali, 2001]. This analysis shows that mean swirl may significantly modify the acoustic and vortical spectral composition of the propagating modes in the duct. The analysis also shows that vortical nearly convected modes have very small pressure content. However, significant vorticity may be associated with the acoustic modes, if the mean ft>w is rotational.

Another important element in the formulation and implementation of the scattering problem is the derivation of nonreflecting boundary conditions. The authors [Ali et al., 2001] have previously derived nonrefecting boundary con­ditions for ‘acoustic’ and ‘vortical’ waves propagating in a duct with swirling ft>w. These boundary conditions have been implemented in an explicit scheme and tested by computing the propagation of ‘acoustic’ and ‘vorticity’ waves and comparing the solutions with the normal mode analysis. These solutions have also been validated in Atassi and Ali, 2002 for the scattering of vortical waves by an annular cascade in a uniform flow by comparison with the integral solutions of Namba and Schulten, 2000.

In § 2, the mathematical formulation is presented with the infow/outlbw boundary conditions. In § 3, numerical results are presented and show how swirl modifies the scattering phenomena and what conditions result in strong scattering of the incident field.