Thierry M. Faure

The ongoing development of low order methods providing fast estimations of aerodynamic loading are essential for pre-dimensioning and optimization of wings and fins. The aim of this study is to present an efficient 3D Non-Linear Lifting-Line Method enhanced by an artificial viscosity correction, able to give reliable results for post-stall regime, where conventional lifting-line methods usually fail. In the present method, the lifting-line governing equation is solved by a Newton-Raphson algorithm, with an analytical Jacobian matrix. An artificial viscosity term has been added to the governing equation, in order to regularize the solution for post-stall regime. A unique parameter was defined to control the amount of artificial viscosity introduced in the governing equation. The investigations focus on the influence of the amount of artificial viscosity on the regularization of the final solution. The perturbation generated by the artificial viscosity on the governing equation is also analyzed. The method has been validated and evaluated with linear and non-linear test cases, from analytical and experimental data from the literature. It was shown that there is an optimal amount of artificial viscosity leading to a converged solution, for post-stall regime, associated with smooth circulations along the wing span, coherent 3D lift coefficients, and low deviations from the initial governing equation. When the amount of artificial viscosity increases beyond this optimal value, the solution deteriorates. The optimal value of the parameter controlling the amount of artificial viscosity has been found to be practically insensitive to the angle of attack, and to the wing discretization.

Cetacean propulsion by a periodic flapping motion of their fluke is considered and studied on a benchmark flexible straight wing. The aim of the study was to validate low-order models for this configuration. First, the two-dimensional rigid case is investigated, comparing the aerodynamic performance of the airfoil periodic motion vs the reduced frequency, with published data and unsteady Reynolds-averaged numerical simulation results. It appears that viscous drag modeling must be added to the discrete vortex method, in order to obtain sensible thrust results, for Garrick frequencies below 2. All high- and low-order models agree at the remarkable Garrick frequency of 1.82, although the experiment shows a lower efficiency of about 25%. The positions of the shed vortices match comparing the unsteady Reynolds-averaged numerical simulation and the discrete vortex method. Then, the three-dimensional leading-edge-suction-parameter modulated discrete vortex method is extended, by means of a lifting line theory. A modification of the method is proposed in order to consider wing dihedral, resulting from the spanwise flexibility. The configuration considers a reduced frequency of 1.82. Three types of spanwise wing flexibility are examined. For the inflexible and flexible cases, a reasonable agreement is observed between the different methods for each coefficient. The intermediate flexible wing provides a better thrust coefficient, while excessive flexibility proves to be detrimental. Vorticity fields are compared with previously published data for the three wings. For the highly flexible wing and the right choice of deformation parameters, the discrete vortex method produces reliable results.

The leading-edge-suction-parameter modulated discrete vortex method is extended to a wing with a finite span and no sweep, in order to get the development of aerodynamic coefficients with an angle-of-attack, from attached to completely detached flow conditions. A first case considering the unsteady pitching motion of a flat plate is compared with published experimental and numerical results. Then, dependence of lift, drag, and pitching moment coefficients with the angle-of-attack is discussed for a wing built on an SD7003 airfoil at a constant angle-of-attack. The three-dimensional effects on the lift coefficient curve for a completely detached wing are established.

When an aircraft is locked in a stable equilibrium at high angle-of-attack, we have to do with the so-called deep stall which is a very dangerous situation. Airplanes with T-tail are mainly concerned with this phenomenon since the wake of the main wing flows over the horizontal tail and renders it ineffective but other aircrafts such as fighters can also be affected. First the phase portrait and bifurcation diagram are determined and characterized (with three equilibria in a deep stall prone configuration). It allows to diagnose the configurations of aircrafts susceptible to deep stall and also to point out the different types of time evolutions. Several techniques are used in order to determine the basin of attraction of the stable equilibrium at high angle-of-attack. They are based on the calculation of the stable manifold of the saddle-point equilibrium at medium angle-of-attack. Then several ways are explored in order to try to recover from deep stall. They exploits static features (such as curves of pitching moment versus angle-of-attack for full pitch down and full pitch up elevators) or dynamic aspects (excitation of the eigenmodes and improvement of the aerodynamic efficiency of the tail). Finally, some properties of a deep stall prone aircraft are pointed out and some control tools are also implemented. We try also to apply this mathematical results in a concrete situation by taking into account the captors specificities or by estimating the relevant variables thanks to other available information.