Analytical formulation of 2D aeroelastic model in weak ground effect

This paper deals with the aeroelastic modeling and analysis of a 2D oscillating airfoil in ground effect, elastically constrained by linear and torsional springs and immersed in an uncompressible potential flow (typical section) at a finite distance from the ground. This work aims to extend Theodorsen theory, valid in a unbounded flow domain, to the case of weak ground effect (GE), i.e., for clearances above half the airfoil chord. The key point is the determination of the aerodynamic loads both in the time and frequency domain involving their dependence on the ground distance. The method of images for ensuring impermeability condition on the ground and asymptotic expansions in the perturbation parameter (defined as the inverse of the non-dimensional ground clearance of the airfoil) are employed for obtaining the vortex distributions along the chord and the wake. The representation of the aeroelastic system is transformed from the frequency domain into the time domain and then in a pure differential form using a finite-state aerodynamic approximation (augmented states). The developed theory is applied to a typical section obtained as a reduced model of a wing box finite element representation, thus allowing comparison with the corresponding aeroelastic analysis carried out within the a commercial solver based on a 3D lifting surface aerodynamic model. The stability (flutter margins) and the response of the airfoil both in frequency and time domain are then investigated. In particular, within the developed theory, the solution of the Wagner problem can be directly achieved confirming an asymptotic trend of the aerodynamic coefficients toward the steady-state conditions different from those relative to the unbounded domain case. The dependence with respect to the ground clearance of the flutter speed and of the frequency response functions is highlighted, showing the usefulness of this approach in efficiently and robustly accounting for the presence of the ground when unsteady analysis of elastic lifting surfaces in weak ground effect is required.