MULTISTAGE COUPLING FOR UNSTEADY FLOWS IN TURBOMACHINERY

Kenneth C. Hall, Kivanc Ekici and Dmytro M. Voytovych

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708-0300

Abstract In this paper, we present the a three-dimensional time-linearized unsteady

Euler solver for computing unsteady flaws in multistage turbomachines. Using this approach, each blade row is modeled with a computational grid spanning a single blade passage. Within each blade passage, several time-linearized un­steady solutions are computed, one for each of several “spinning modes” re­tained in the model. Each spinning mode has its own frequency and interblade phase angle. These various solutions are coupled together at the inter-row bound­aries between the blade rows. Results are presented for several geometries, and demonstrate the accuracy and efficiency of the method, as well as the impor­tance of multistage effects. In particular, we show that multistage effects can strongly affect the aerodynamic loads acting on a given blade row. Further­more, the method presented is highly efficient. For example, to perform a flitter calculation requires only about three times the CPU time of one steady flaw computation for each interblade phase angle and frequency considered.

1. Introduction

Unsteady fhid motion is essential to gas turbine engine operation. Only through unsteady ft>w processes can a machine do work on a fliid to increase its total enthalpy. This unsteadiness is provided in compressors and turbines by relative motion of adjacent stators and rotors. Unfortunately, this motion also produces undesirable aeroacoustic and aeroelastic phenomena, i. e. tonal noise and forced blade vibrations induced by rotor/stator interactions. Furthermore, the aeroelastic (flitter) stability of a rotor can be profoundly inflienced by the presence of nearby stators and rotors.

Most current unsteady aerodynamic theories model a single blade row in an infinitely long duct, ignoring potentially important multistage effects. How­ever, unsteady flows are made up of acoustic, vortical, and entropic waves.

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K. C. Hall et al. (eds.),

Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 217-229. © 2006 Springer. Printed in the Netherlands.

These waves provide a mechanism for the rotors and stators of multistage ma­chines to communicate with one another.

Recently, a number of investigators have begun to examine the importance of multistage effects on flitter and forced response of cascades, see e. g. Buf – fum, 1993, Hall and Silkowski, 1997, Silkowski, 1996, and Silkowski and Hall, 1998. Silkowski & Hall have developed a ‘Coupled Mode’ analysis, an elegant and computationally efficient method for modelling neighboring blade row ef­fects in realistic two-dimensional compressors. Using this approach, the cou­pling between blade rows is modelled using a subset of the so-called ‘spinning modes,’ pressure and vorticity waves that propagate between the blade rows. The blade rows themselves are represented by reflection and transmission co­efficients. These coefficients describe how spinning modes interact with, and are scattered by, a given blade row. The coefficients can be calculated us­ing any standard isolated blade row model. In particular, Hall & Silkowski used a linearized full potential fbw model together with rapid distortion the­ory to account for incident vortical gusts. The isolated blade row reflection and transmission coefficients, inter-row coupling relationships, and appropri­ate boundary conditions are all assembled into a small sparse linear system of equations that describes the unsteady fl>w in a multistage machine arising from a prescribed blade vibration of one of the blade rows.

Using the Coupled Mode analysis, two important observation were made. First, the aerodynamic damping of a blade row that is part of a multistage ma­chine will, in general, be significantly different than that predicted using an isolated blade row model. This is an important result since virtually all un­steady aerodynamic theories currently used in industry assume that the blade row can be modelled as isolated in an infinitely long duct. Second, a good estimate of the aerodynamic damping can be obtained using just a few spin­ning modes in the model. In fact, most of the unsteady aerodynamic coupling between blade rows occurs in the fundamental spinning mode, that is, the spin­ning mode associated with the original disturbance. Scattered modes are rela­tively less important.