Mid-term needs associated to aircraft engine design are focused on raising safety and performance while reducing weight, fuel consumption and noise emissions. In order to guarantee the safe operation of the engine, the prediction of the blade aerodynamic stability is essential. For that purpose, unsteady linearized CFD methods based on small disturbances about the nonlinear steady-state system behaviour are commonly used due to their reasonable accuracy at acceptable computational costs. However, for moderate blade oscillation amplitudes, nonlinear aerodynamic effects may appear, yielding eventually limited, finite oscillation amplitudes and an evolution into a stable, limit cycle oscillation (LCO) instead of flutter. As a consequence, linear methods may be overly restrictive for the blade design.

In order to determine the blade stability at moderate amplitude values, nonlinear unsteady CFD methods can be used. However, the direct time integration of the URANS equations has shown to be very expensive, as several periods have to be computed in order for the transient behavior to vanish. The harmonic balance (HB) method takes advantage of the turbomachinery unsteady periodic flow features by expanding the conservative variables of the RANS equations in a Fourier series for a finite number of integer multiples of the fundamental harmonic. It explicitly neglects this transient behavior and thus provides an efficient tool to determine the nonlinear aerodynamic response, the compromise thereby being the consideration of a limited number of higher harmonics only, which in turn depends on the severity of the aerodynamic nonlinearity under consideration. Hence, the challenge lies in the appropriate selection of the number of harmonics to be retained in the nonlinear computation. To this end, the prediction of the nonlinear system behavior is typically verified using either a convergence study with an increasing number of harmonics, or a direct comparison with expensive time-domain simulations.

This work proposes a simple and systematic approach to determine the number of harmonics to be retained when using the HB method. First, the method is applied to the well-known Duffing oscillator. Next, a compressor rotor blade is investigated for flow conditions with and without aerodynamic nonlinearities. By means of the prescribed motion approach, the blade motion is limited to the fundamental harmonic whereas several harmonics are considered in the aerodynamic forces acting on the blade. Results are validated with experimental data.

The present method verifies the harmonic convergence without any additional, comparative simulations. It can be readily extended to multistage configurations and to a full aeroelastic HB method retaining several harmonics in the blade motion.


Applicability of Harmonic Balance for the Determination of Blade Stability Within the Prescribed Motion Approach

Division / Sponsor:
International Gas Turbine Institute (IGTI)

Paper Proceedings: https://asmedigitalcollection.asme.org/GT/GT2020/volume/84058

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