Topological Interpretation of Non-Linear Transonic Aeroelastic Phenomena

Recent computational studies based on numerical solutions of the unsteady Euler equations have revealed hitherto unanticipated transonic aeroelastic phenomena characterized by non-linear flutter and divergence and interactions between divergence and flutter. In the absence of extensive parametric searches, however, such quantitative studies provide little insight into the nature and extent of such bifurcational behaviour, particularly when multiple parameters are involved. Qualitative dynamical systems theory offers a complementary approach to the analysis of bifurcation problems. In the vicinity of bifurcation, the qualitative behaviour of complex dynamical systems can often be characterized by simple ordinary differential equation models. Of particular interest are the simplest models which exhibit complex interactions characteristic of those observed in non-linear aeroelastic systems. Such models offer scope for attaining greater insight into the nature of complex aeroelastic bifurcations and for systematically predicting qualitative changes resulting from parameter variations. The present work describes the elements of a qualitative, or topological, model identification strategy for a general class of aerodynamically non-linear hereditary aeroelastic systems. The methodology is motivated, principally, by a desire to circumvent the need for detailed knowledge of the unsteady aerodynamic environment. The approach employs a notional non-linear functional description of the aerodynamic force response free from any low-frequency or quasisteady aerodynamic assumptions. Application of the scheme to the transonic aeroelastic problem demonstrates the feasibility and limitations of qualitative techniques. Based on partial bifurcational information derived from published numerical solutions of the coupled aerodynamic and structural equations of motion, a simplified model is identified which captures the (local) bifurcational behaviour of a structurally linear typical section aerofoil in 2-D transonic flow. The model facilitates a new interpretation of divergence/flutter interaction phenomena in transonic flow, including the effects of structural asymmetry, and highlights some difficulties of definition and interpretation of non-linear flutter. Evidence is presented which suggests the existence of new transonic aeroelastic phenomena not previously encountered in computational studies