Investigation of fluid-structure interaction in vibrating cascades using a time domain method

Kurzfassung

The development of modern aircraft engine compressors with increased pressure ratio and reduced weight has led to hightly loaded stages with transonic inflow. As a result, the design engineers of completed engines have often encountered severe aeroelastic problems, the solution of which could be difficult and costly. Therefore, tools are needed which already in the design process are able to correctly predict the aeroelastic behaviour of the blading. In the classical linearized approach the aeroelastic problem is divided into separate steps of aerodynamic and structural computations. Assuming a linear dependence of the structural and aerodynamic forces on the local deflections, the mode shapes of the structural dynamic model are used to generate harmonic motions for the calculation of airloads. In a second step the aeroelastic problem is handled by using these airloads in the governing structural equations, which are solved in the frequency domain. This requires, e.g., the solution of an eigenvalue problem, if the aeroelastic stability boundary has to be determined. However, nonlinear fluid-structure interaction caused by oscillating shocks or strong flow separation may significantly influence the aerodynamic damping and hence effect a shift of flutter boundaries. In order to investigate such aeroelastic phenomena, the governing equations of structural and fluid motion have to be simultaneaously integrated in time. The various techniques for integrating the structural together with the flow equations can be roughly divided into two methods: Time-staggered algorithms advance the structural and fluid system at subsequent time steps leading to a small time lag between the integration of structure and fluid. In contrast to this, fully coupled algorithms make use of iterative solvers to update both the structural and fluid system at the same time level. Although staggered algorithms are always leading to small errors in energy conservation, their greatest advantage is their ability to use well established discretizations and solution methods in each of the two disciplines. In this paper a technique is presented which analyzes the aeroelastic behaviour of an oscillating compressor cascade by a time-staggered method. The structural part of the governing aeroelastic equations is time-integrated according to the algorithm of Newmark, while the unsteady airloads are computed at every time step by an Euler code. The link between the two time integrations is an automatic grid generation in which the used mesh is dynamically deformed as such that it conforms with the deflected blades at every time step. The capability of this method to predict aeroelastic stability is demonstrated for subsonic and transonic flow. In order to concentrate the investigations on the principal physical effects which dominate the fluid-structure interaction in sub- and transonic flow, a very simple structural model was taken as a basis for the computations. This model consists of an assembly of twenty compressor blades performing torsional vibrations around midchord. The structural dynamic properties of each of the blades were simulated by a mass-spring-damper system performing a one-degree-of-freedom vibration under the influence of an aerodynamic moment which in general is a nonlinear function of the torsional deflection. Time series were computed for tuned and mistuned blade assemblies operating in subsonic and transonic flow. It was found that for subsonic flow the differences between time domain and frequency domain results are of negligible order. For transonic flow, however, where vibrating shocks and a temporarily choked flow in the blade channel dominate the unsteady flow, the energy transfer between fluid and structure is no longer comparable to that of a linear system. This is demonstrated by Fig.1 showing the Iso-Machlines in adjacent blade channels 70 milliseconds after the start of the computation. The time series of the blade deflections resulting from the unsteady flow of Fig.1 are depicted in Fig.2 showing the pitching angles of four adjacent blades. A salient feature of this figure is the vibration of blades with odd and even numbers around different equilibrium positions and the lack of a clearly discernible traveling wave flutter mode. The results of the transonic flutter case depicted in Figures 1 and 2 clearly demonstrate that here the application of the time domain method leads to a significantly different aeroelastic behaviour of the blade assembly including a shift of the stability boundary.