In this work limit-cycle oscillations (LCOs) caused by aerodynamic non-linearities were analysed using fluid-structure interaction (FSI) simulations of a two-degree-of-freedom airfoil system. A comparison with sinusoidal forced motion oscillation simulations at the fundamental frequency was performed. The results show that using the first harmonic component for the forced motion oscillations is a good approach to describe the limit-cycle oscillations considered in this work. Following the work of G. Dietz, G. Schewe and H. Mai (2004), Journal of Fluids and Structures, Vol. 19, an energy budget analysis of the limit-cycle oscillations was performed. Power analysis of the LCO reveals that the power of the lift and moment show non-linear behaviour with increasing amplitude. The linearised equivalent power components that would occur in case of flutter were computed. These show that the defect in the power of the lift in the non-linear case is caused by the increase of the phase of the lift with oscillation amplitude, which is the result of the unsteady shock wave motions on both upper and lower surface of the airfoil. The power of the moment also shows a defect, which is much smaller than in case of the lift. This defect is caused by variations in both the magnitude and the phase of the moment with oscillation amplitude.

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Aeroelastic limit-cycle oscillations (LCO) due to aerodynamic non-linearities are usually investigated using coupled fluid-structure interaction simulations in the time domain. These simulations are computationally expensive, especially if a large number of LCO solutions must be computed to study the Hopf bifurcation behaviour in the immediate surrounding of the flutter point. To facilitate such bifurcation parameter studies an adaptation of the well-known p-k flutter analysis method is proposed in this paper. In this frequency domain method, the first harmonic of the motion-induced unsteady aerodynamic forces is no longer assumed to be solely determined by constantcoefficient frequency response functions. Instead, the non-linear dependence on the oscillation amplitudes and the phase angle between the input degrees of freedom is additionally taken into account. Therefore, the first harmonic Fourier components of the aerodynamic forces are sampled and interpolated in advance. The LCO solution is then found iteratively. The proposed amplitude-dependent p-k method (ADePK) is applied to a classic two-degree-of-freedom spring-mounted airfoil system where the non-linear aerodynamic forces are computed from Euler simulations. Fluid-structure

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Flutter is usually predicted using linearised theory. In reality, flutter is always non-linear and might already occur below the linearly predicted flutter boundary. Whether this is the case for limit-cycle oscillations (LCOs) caused by aerodynamic non-linearities is not known, since these LCOs can only be predicted using expensive wind-tunnel tests or coupled Computational Fluid Dynamics (CFD)-Computational Structural Mechanics (CSM) simulations. However, it is important to know whether a sufficiently large disturbance can already cause LCOs below the flutter boundary predicted from linearised theory. Furthermore, since structural properties and the flow conditions will vary, it is necessary to study the resulting variations of the Hopf bifurcation behaviour of the LCO solutions near the flutter point. In this work viscous and inviscid transonic flows are considered. The LCO bifurcation behaviour was found to vary significantly when the uncoupled structural natural frequency ratio and the location of the elastic axis are changed. When the non-linearity is relatively weak, a change in the Hopf bifurcation type might result. A Mach number variation in inviscid flow showed that the effective flutter boundary might significantly deviate from that predicted using linearised theory. For both the structural parameter variations and the Mach number variation, LCOs were observed below the linearly predicted flutter boundary. At the nominal structural parameters, the amplitude-dependent behaviour of the phase of the lift was found to be responsible for the type of bifurcation of the LCO solution that occurs. Inspection of the local force distributions at various pitch amplitudes showed that the motion of the shock wave on the lower surface is responsible for the behaviour of the phase of the lift and hence for the bifurcation behaviour of the LCOs observed in this work.