FREQUENCY AND TIME DOMAIN FLUID-STRUCTURE COUPLING METHODS FOR TURBOMACHINERIES

Duc-Minh Tran and Cedric Liauzun

Structural Dynamics and Coupled Systems Department Office National d’Etudes et de Recherches Aerospatiales B. P. 72, 29 avenue de la Division Leclerc, 92322 Chatillon Cedex, France minh. tran_duc@onera. fr, cedric. liauzun@onera. fr

Abstract Two methods of fUid-structure coupling for turbomachinery are presented, the first one in the frequency domain and the second in both frequency and time domains, with the assumptions of linearized aerodynamics and cyclic symmetry.

Keywords: fliid-structure coupling, aeroelasticity, turbomachinery

1. Introduction

This paper is concerned with the coupled fliid-structure dynamic analysis of turbomachinery. The structure consists of a rotating bladed disk submitted to the unsteady aerodynamic forces exerted by the fliid, which are themselves generated by the structural motion. The structure and the fluid are assumed to have a perfect circumferential cyclic symmetry, so that the classical reduc­tion of the analysis to only one reference sector can be applied. The study of the structure comes down to that of the reference sector by applying the ap­propriate boundary conditions for each phase number. The displacements of the reference sector in the travelling wave coordinates are expressed as a lin­ear combination of the complex modes or Craig and Bampton’s basis and the motion equations are projected on these bases to obtain a reduced system.

In the coupling methods proposed here Tran et al., 2003, the unsteady aero­dynamic forces are assumed to depend linearly on the structural displacements and velocities and they are expressed in terms of those induced by the modes. The mode-induced aerodynamic forces are computed only once at the begin­ning of the simulation by using an aerodynamic code (solving the Euler equa­tions) with the assumption of harmonic motion of the modes, for an inter-blade phase angle and a number of reduced oscillation frequencies.

397

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

Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 397-408. © 2006 Springer. Printed in the Netherlands.

In the frequency domain, the projection of the mode-induced aerodynamic forces on the modes provides a complex matrix of aerodynamic coefficients whose product with the modal coordinates gives the generalized aerodynamic forces, leading to a nonlinear eigenvalue system. This flitter equation is solved by using two well-known iterative techniques already used for aircrafts, which are the double scanning method Dat and Meurzec, 1969 (also called p — k method) and Karpel’s minimum state smoothing method Karpel, 1982.

In the time domain, Karpel’s minimum state smoothing of the aerodynamic coefficient matrix is used to obtain a time-domain approximation of the gen­eralized aerodynamic forces by means of auxiliary state variables. Structural nonlinearities such as friction or free-play are taken into account. This nu­merical strategy is applied to a compressor blade for different configurations (phase numbers, rotation speeds). The results are compared between the pro­posed methods and to those of the direct coupling method Sayma et al., 2000 where the structural and the fluid equations are solved alternately at each time step and the assumptions of linearity and harmonic motion are not necessary.